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Article Text

Tobacco industry pricing strategies for single cigarettes and multistick packs after excise tax increases in Colombia
1. Zaineb Danish Sheikh1,
2. J Robert Branston1,2,
3. Blanca Amalia Llorente3,
5. Anna B Gilmore1
1. 1Tobacco Control Research Group, Department of Health, University of Bath, Bath, UK
2. 2School of Management, University of Bath, Bath, UK
3. 3Fundación Anáas, Bogotá, Colombia
4. 4PROESA - Research Center on Health Economics and Social Protection, Department of Economics, Universidad Icesi, Cali, Colombia
1. Correspondence to Dr J Robert Branston, School of Management, University of Bath, Bath BA2 7AY, UK; J.R.Branston{at}bath.ac.uk

## Abstract

Introduction Taxes on tobacco products are an efficient way of reducing consumption. However, they are only effective if passed on to consumers with higher prices. This study aims to examine tobacco industry (TI) pricing strategies in response to tax increases, and whether they differ by price segments or presentation (packs or individual sticks) in Colombia. This is the first such academic study in Latin America and the first anywhere to include the market for single sticks.

Methods Using data on cigarette pricing/taxation from a survey of smokers (2016–2017) and official government data on pricing (2007–2019), the TI’s pricing strategies were examined, split by brand, price segments, different sized packs and single cigarettes.

Results The TI employed targeted pricing strategies in Colombia: differentially shifting taxes; and launching new brands/brand variants. The industry overshifted taxes when increases were smaller and predictable, but used undershifting more when there was a larger increase in 2017, after which it mostly overshifted on budget and premium (but undershifted mid-priced) brands. The prices for single sticks increased more than the tax increase in 2017 when their consumption also increased.

Conclusion The pricing strategies identified suggest excise taxes can be increased further, particularly the specific component, to reduce the price gap between brand segments. Brands should be restricted to a single variant along with prohibitions on launching new brands/brand variants. Lastly, since the pricing of single sticks does not match the pattern of packs, more monitoring of their sales and distribution is required, especially since they promote consumption and hinder effective implementation of tobacco tax policies.

• economics
• low/middle income country
• price
• taxation
• tobacco industry

## Data availability statement

The DANE data are freely available from the listed online sources. The DEICS-COL survey data can be shared with researchers who meet the relevant criteria for access—help and information can be obtained directly from Fundación Anáas.

This is an open access article distributed in accordance with the Creative Commons Attribution 4.0 Unported (CC BY 4.0) license, which permits others to copy, redistribute, remix, transform and build upon this work for any purpose, provided the original work is properly cited, a link to the licence is given, and indication of whether changes were made. See: https://creativecommons.org/licenses/by/4.0/.

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#### WHAT IS ALREADY KNOWN ON THIS TOPIC

• Tobacco companies use price-based strategies in high-income countries to minimise the impact of taxation, but it is unclear what they do in low and middle-income countries (LMIC), particularly with single sticks.

• We explore how cigarette prices in Colombia changed between 2007 and 2019, covering both packs and single sticks.

• Taxes were differentially shifted for different price tiers of cigarettes and with the quantity of cigarettes purchased.

#### HOW THIS STUDY MIGHT AFFECT RESEARCH, PRACTICE AND/OR POLICY

• The results suggest more work is needed to understand and address tobacco pricing in LMIC contexts, particularly of single cigarettes.

## Introduction

Higher prices as a consequence of increased tobacco tax are one of the most effective and cost-effective measures available to curb the tobacco epidemic. It has an even greater effect on reducing consumption in low and middle-income countries (LMIC) and among youth.1–4 The WHO Framework Convention for Tobacco Control (FCTC) calls for higher taxes and prices for tobacco products in order to reduce tobacco-related morbidity and mortality.5 However, it is a relatively underused tobacco control measure,6 7 with, as of 2021, only 13% of the world’s population living in the 40 countries that meet the WHO’s recommendation of taxing at 75% or greater of the retail price of the most popular cigarette brands.8–10

Colombia is one such country where excise taxes on tobacco products fall short of the 75% benchmark.9 It scored 3.38 out of 5 on the 2021 Tobacconomics cigarette tax scorecard which assesses countries’ cigarette tax policies in relation to widely accepted best practices.11 This score represents a considerable increase on the 2016 score, 2.38,12 but shows there is still room for substantial improvement, particularly in cigarette prices and the tax share of price. Indeed, despite several reforms with limited tax increases between 1997 and 2010,13 taxation on tobacco remained one of the lowest in Latin America in 2016, while the smoking prevalence was one of the highest at 32.2%, particularly among children aged 13–15 years.14–17 In 2016, the nominal price of 20 Marlboro cigarettes was 3872 Colombian pesos (COP$) (then US$0.97).18 19

In 2017, as part of a larger fiscal reform, a major excise tax increase on tobacco products was introduced along with an increase in the general value-added tax (VAT).20 The reform doubled the specific component of the excise tax from COP$700 (US$0.23) to COP$1400 (US$0.47) per 20-stick pack (the ad valorem tax remained unchanged at 10% of the retail price charged to the public), while VAT was increased from 16% to 19% of the base price. The specific tax increased to COP$2100 (US$0.74) in 2018, and from 2019, an annual tax escalator was implemented where the specific tax is increased by the country’s annual rate of inflation plus four percentage points each year.21 22 These changes in tax were expected to substantially decrease tobacco consumption by increasing the retail price of tobacco. However, retail prices are established by the tobacco industry (TI) who do not have to pass on the increased taxes (as it may not align with their profit-making strategies) and hence can (somewhat) mitigate the impact on consumption. In Colombia, the market is essentially contested between the two transnationals, British American Tobacco (BAT; 55% market share according to Euromonitor) and Philip Morris International (PMI; 43%),23 giving each considerable market, and hence pricing power. Previous research highlights that the TI employs a variety of pricing strategies to undermine tax increases in many countries.24–26

TI pricing tactics are an area of increasing concern worldwide as they weaken the effectiveness of tobacco tax policies. However, relatively few studies have examined the TI’s price-based responses to taxation, with the majority from high-income countries (HIC)27–30 such as the UK31–34 and USA,35–41 and a smaller number exploring LMICs.24 In particular, six broad pricing strategies have been identified that are consistently used across different countries: differential tax shifting (both overshifting and undershifting); launching new brand variants/products; product promotions; smoothing prices after a tax increase (ie, avoiding quit-inducing jump in prices via smaller, incremental more frequent adjustments to prices); disguising price increases by reducing the number of sticks per pack (where legally allowed); and changing product attributes or production processes.24 31 These TI approaches have not been prominent in taxation discussions in LMICs, where tobacco taxes are relatively low compared with HICs and cigarette affordability is often increasing as a result of income growth. In addition, one particular aspect of concern regarding industry’s price-based responses to taxation that has been ignored thus far is around the pricing of single cigarette sticks which is a significant issue in LMICs (although not necessarily entirely controlled by the TI). By obviating the need to buy a pack, the sale of single sticks makes tobacco more affordable, hindering effective taxation policies42 while also providing distribution channels for illicit cigarettes. It might therefore provide the TI with an additional avenue to react to, and undermine, taxation.43–45 Although it is illegal to sell loose cigarettes in Colombia since it adopted the FCTC’s Article 16 in 2009 (which prohibits the sale of loose cigarettes or in small packages), they continue to be sold, especially by a large volume of street vendors. Moreover, there is some anecdotal evidence from Colombia that the TI works closely with retailers, and that they take advantage of and encourage the informal market (where single sticks are sold). Consequently, Colombia represents an interesting market to explore the TI’s price-based responses to tax increases with potentially many lessons to learn, especially since no academic study to date (that we are aware of) has considered a Latin American country.

The aim of this study is therefore to address these gaps in knowledge by exploring tobacco pricing in Colombia, including single sticks. To triangulate our analysis, we use two data sources: a survey of the smokers (2016–2017) and official government data on prices (2007–2019). Given the limited studies on the TI’s tax pass-through in LMICs and the lack of research on pricing of single sticks (despite their growing rate of sales in multiple countries),46–48 this study will be relevant globally.

## Methods

### Data sources

The first data source analysed is ‘the Demand for Illicit Cigarettes Survey for Colombia’ (DEICS-COL), which is a nationally representative cross-sectional survey of smokers aged 12–65 years carried out in two waves (2016 and 2017) by Fundación Anáas (an independent civil society organisation promoting public health). The 2016 survey was carried out 4 months before a major tax increase in Colombia while the 2017 wave was collected 8 months after. Details of the study enrolment and protocols have been described previously.43 49 50 The interviewer-administered survey involved a mix of two methods: smoker’s self-report on consumption pattern, brand and last purchase information including pack size, price paid and place of purchase; and interviewer’s direct observation of cigarette packs/sticks to validate self-reporting. Each wave contained 1697 respondents.

The second source is national-level average cigarette prices issued twice annually (semester I in December and semester II in June each year) by the National Administrative Department of Statistics (DANE). DANE publishes, for the purposes of identifying the price levels on which the excise duties due, the retail prices for all cigarette brands/brand variants (henceforth brands for simplicity) available in supermarkets,18 which represents approximately half of the total market. (The remainder of the market is dominated by the informal sector where retail prices will vary more.) For simplicity and consistency, we analysed the semester II data for each year between 2007 and 2019.

### Analysis

We explored cigarette pricing from both data sources separately to understand whether tax changes were entirely translated into price changes or they were shifted differentially, that is, undershifted or overshifted or both. We also explored changes to the brands that were available/used. Since our data cover retail prices, we are unable to distinguish between the wholesale pricing behaviours of the TI and any impact that retailers/distributors had with their own pricing decisions. Furthermore, for simplicity, we treated all sales of loose sticks as if they were duty-paid sales because a significant part of this market is the resale of legal, duty-paid cigarettes bought in multistick packs, and which would therefore be impacted by the tax change. Moreover, such an impact would likely also affect the selling price of illicit products via such channels. All prices are reported in COP and were adjusted to real prices by removing the impact of inflation using the World Bank’s consumer price index51 measure of inflation in Colombia, with 2017 chosen as the base year (as the second year of the survey).

#### DEICS-COL survey

Univariate descriptive analysis was used to characterise trends in cigarette prices, tax changes and, hence, net prices (the industry’s earnings from sales once all taxes have been paid). The analyses were conducted using SPSS V.26. Packs were categorised by their number of cigarettes (10, 14, 18 and 20). Cigarette brands were segmented into three hierarchical price categories (economy, mid-priced and premium) by the authors based on the price points in the market over the whole period of the study, following the WHO approach of weighted-average price tertiles.52 53

In order to compare packs of different sizes we used price per stick as the key measure. Where it was not provided in the data, we calculated price per stick by dividing the reported per-pack price by the number of sticks in the pack. The taxes paid per individual cigarette were then calculated for each brand based on its selling price. The net price per stick for each price segment was calculated by subtracting the total tax (excise (specific and ad valorem) plus VAT) from the total price per stick. Tobacco tax pass-through for each segment was calculated by calculating the changes in net price per stick. The percentage change in price attributable to government tax was calculated by dividing the total tax increase by the total price increase and then multiplying by 100, while the percentage change due to TI revenue was calculated by dividing the increase in net revenue by the price increase and then multiplying by 100. We also calculated the frequency of smokers reporting using brands in the different market segments and how these changed between the two survey waves. Such changes were tested for significance using χ2 tests.

#### DANE data

The data from DANE contained information on the price of packs of 20 cigarettes between 2007 and 2019 (based on data availability). The tax paid per pack (excise and VAT) was calculated based on the price for each brand. Average net price per pack for each brand was then calculated by subtracting the total tax paid from the average price. This was used to track how taxes were shifted to price. We calculated changes in net price by subtracting the net price of the previous year from the net price of the current year.

## Background results

The DEICS-COL survey has been used before; we present a summary of it here as appropriate background to enable our results to be put into context. For more details, please see the previous publications,43 49 50 and the online supplemental appendix for a summary table.

### Supplemental material

After the tax increase in 2017, the average self-reported price for cigarettes increased for all price categories (online supplemental table 1) and more smokers purchased cigarettes from street vendors.43 As observed in both years, more smokers bought singles at their last purchase as compared with packs and cartons, and the frequency increased from 61.8% in 2016 to 73% in 2017 which was statistically significant.50 In regard to the sale of packs, a pack of 10 sticks is more commonly purchased as compared with packs of 20, 18 or 14 (a single observation in 2017). After the tax increase, the frequency of purchasing all varieties of packs also decreased by 11% (significant at 1% confidence level).43

## Results

### Comparison of DEICS-COL survey prices with DANE prices

A comparison of the survey prices with those from DANE for brands that were present in both the sources revealed that they were broadly consistent. However, the survey prices were generally slightly higher than DANE’s, and these variations were more noticeable in 2017 (figure 1).

Figure 1

Consistency of Demand for Illicit Cigarettes Survey for Colombia (DEICS-COL) survey retail prices with National Administrative Department of Statistics (DANE) prices for packs of 20 (2016–2017). Source: Authors’ own calculations using the database of DANE and DEICS-COL survey.

### DEICS-COL survey

#### Tax pass-through analysis

An analysis of the changes in the retail price and tax component of survey data between 2016 and 2017 (table 1) showed that for loose sticks the price for all segments increased more than the tax increase, with economy having a greater relative increase in net price followed by mid-priced and premium brands. Similarly, economy packs of 10 and 20 cigarettes were also slightly overshifted. However, taxes on mid-priced and premium packs of 10, 18 and 20 sticks were absorbed to some extent by the industry with their net prices decreasing during the year, signifying undershifting of taxes. These results suggest that the TI has differentially shifted taxes during 2017 for different presentations and pack sizes of cigarettes.

Table 1

Changes in the real price and tax for different presentation of cigarettes between 2016 and 2017 (prices per stick; all monetary figures in COP)

The unit cost of a loose stick is almost double that of a stick purchased in packs of 10, 18 or 20 sticks (figure 2). For cigarettes bought in packs, the highest priced were sold as packs of 10, and the lowest in packs of 18 cigarettes.

Figure 2

Changes in median price per stick for different presentation of cigarettes, 2016–2017. Source: Authors’ own calculations using the data from the Demand for Illicit Cigarettes Survey for Colombia (DEICS-COL) survey. VAT, value-added tax.

An analysis of the popularity of the different market segments (online supplemental table 2) showed that mid-priced cigarettes were most popular in 2016 (51%), but in 2017 it was premium (50%). The frequency of mid-priced brands decreased between the 2 years with a statistically significant shift to economy and premium brands. Additionally, taxes on some brands were more absorbed than others in the same price categories (online supplemental table 3). For example, Marlboro was more undershifted than Lucky Strike (premium), Belmont more than Mustang (mid-price) and D&J more than Starlite (economy). The reasons for this are unclear but it was consistently PMI brands that were more undershifted, so it likely relates to the different profit-maximising strategies of the tobacco companies and the competition that exists between their brands.

### DANE data

#### Descriptive

DANE data analysis revealed that between 2007 and 2019, the industry launched new variants of several existing brands (L&M, Lucky Strike, Marlboro), and this happened throughout the period under analysis, particularly during the most recent years. Furthermore, international brands Chesterfield and Rothmans were introduced as replacements for existing brands (Boston and Mustang, respectively) in 2017–2018 (packs featured both names for a time). Similarly, other new flavour variants with mint/menthol, fruit and/or beverage flavourings were also launched (eg, Lucky Strike Mojito) as were strength variants (eg, L&M Red).

##### Tax pass-through analysis

Examination of the changes in the price and tax components of DANE data revealed that the TI increased prices beyond the tax increase for most brands between 2007 and 2016, although there were a few exceptions in different price categories throughout this period (table 2 and online supplemental table 4). The major tax increase in 2017 brought about a change in the industry pricing strategy from overshifting to undershifting, as seen by the drop in net prices for all brands. However, the price changes between brands that were withdrawn, and new replacement brands/brand variants showed that the prices were notably increased for both Rothmans and Chesterfield beyond that required by the tax increase alone (change in net price between COP$392 and COP$541). Overall, the tax changes in 2018 and 2019 were largely overshifted except for a few mid-priced varieties, including prices for brands that were replaced in the market (Mustang and Boston).

Table 2

Year-to-year change in real net price for different brands and brand variants of cigarettes between 2008 and 2019 (adjusted to 2017 COP)

## Discussion

This study provides a comprehensive overview of the TI’s pricing strategies in Colombia, including single-stick sales, and therefore contributes to the evidence base on TI’s tax pass-through to consumers in LMICs. Between 2007 and 2016, while tax increases were small, the industry was consistently overshifting taxes, thereby increasing its profitability. This seemed to change following the large tax increase in 2017, when a more complex pattern emerged where taxes on loose cigarettes and economy segments of packs of 10 and 20 were overshifted while they were partially absorbed on packs of mid-priced and premium. That taxes were not shifted equally between packs and single sticks suggests that the pricing of single sticks moves slightly differently from that for packs. From 2017, there is some disagreement between the DANE pricing information and that identified in the survey of smokers as the former showed a substantial amount of undershifting. This difference could be because DANE only records prices of 20-stick packs in the supermarkets, thereby potentially missing out an important part of the market. Given this discrepancy, it would suggest that DANE needs to broaden the scope of their data to capture the informal market, including single-stick prices. Furthermore, since DANE are reporting prices for some brands that have been withdrawn, it seems as if they might also need to improve their technical accuracy and/or their data collection methodologies. After such improvements, the nature of the data might offer future possibilities for further research exploring causal behaviours in this area.

Whenever the TI overshifts taxation it signifies a missed opportunity for government as the higher prices could have been caused by tax increases instead of enhanced industry margins. Since overshifting has continued to be observed in some brands/segments in the years after the large tax increase in 2017, this implies there is further scope for larger tax increases, especially since there was no meaningful increase in illicit sales following the larger increases from 2017.43

The tax shifting patterns in Colombia are in concordance with other studies on pricing tactics from other parts of the world, where the industry has been shown to either overshift29 33 37 54–56 or undershift57–62 taxes, or practise selective overshifting and undershifting on different price segments. For example, in most HICs,32 34 38 56 and some LMICs,63 64 in response to tax increases, the industry ensures smaller price increases for budget and mid-priced brands while setting relatively higher prices for premium brands. However, in the Colombian cigarette market, taxes on mid-priced (since 2017) and premium (in 2017) brands were largely absorbed which might reflect the different stages of the tobacco epidemic and the particular structure of the markets. Smoking prevalence has reduced in Colombia from 12.9% in 2013 to 8.7% in 2018,65 accompanied by a consumption shift towards smoking fewer sticks per day since 2013.66 Furthermore, like the survey results herein, Euromonitor data suggest there has been a shift away from mid-priced cigarettes in favour of economy and (to a lesser extent) premium brands.67 The pricing of mid-priced brands might therefore be an industry strategy to mitigate these trends away from the middle of the market, and hence a different tactic to demand maximisation that is practised in other LMICs.

The availability of single sticks and their high prevalence among smokers complicates the Colombian market and the TI may be using the informal channels to obscure their pricing tactics, since their prices there are not visible. Furthermore, the existence of such informal channels encourages loose-cigarette sales, thereby weakening the impact of tax increases. From a retailer’s perspective, the profit margins on selling single sticks are substantially more than on packs,68 therefore making compliance with the existing law that bans such sales harder to achieve.48 69 Indeed, the proportion of smokers who bought singles increased after the 2017 tax increase. Such sales may potentially cause a loss of government revenues as it becomes difficult to assess whether taxes have been paid on single sticks (some will have where legal purchases of multistick packs have been resold as single sticks). These findings indicate the weak governance of the national and subnational enforcement authorities who are in charge of regulating and controlling the distribution of cigarettes in the informal market.17 48 70 Considering the structural governance obstacles that perpetuate informal vendors, it is very difficult to eliminate these practices as they require interinstitutional interventions, beyond tobacco control measures. This requires exploring ways to make the tobacco companies accountable for their distribution chain, in combination with interventions directed to retailers that go beyond enforcement.

There was also evidence of the TI using the tactic of introducing new brands/brand variants in the market. This has been observed in other countries such as the UK,32 34 Spain,71 Bangladesh58 and Thailand.72 However, unlike these other markets where variants were often cheaper substitutes to appeal to price-sensitive consumers, in Colombia new variants were often flavour/strength related.34 Colombia is one of 18 countries across the globe that have a large market share for flavoured cigarettes,23 suggested to have exceeded 20% in 2019,73 in line with the DEICS-COL survey results that suggest consumers choose brands based more on flavour and less on the price.49 The introduction of flavours could be related to efforts to attract and maintain smokers in a declining market, and to offer consumers additional features to justify the increasingly higher prices for cigarettes.

### Strengths and limitations

The limitations of this paper relevant to the survey data are the cross-sectional nature of it, which means determination of causality and hence generalisability is unclear. Furthermore, the timing of the survey waves in 2016 and 2017 means they are now several years old and that we were unable to explore the impact of the tax increase beyond the year immediately after its introduction. The survey data were therefore triangulated with government data on all price segments of cigarettes rendering greater confidence in the broad validity of the results. It should, however, be noted that the DANE data cover the prices on which taxes are to be paid, akin to recommended retail prices, and not the actual prices that were actually charged to consumers, which may differ between retailers. In regard to the evaluation of the tax pass-through, we cannot say for certain if their prices were overshifted by the retailers or the TI, and this is especially true with the informal sector selling single sticks where the industry is likely to have less influence. We also treated all loose sticks as if they were duty-paid sales (even though single-stick sales are not legal) but previous work has reported that in 2016/2017, 3.5% of the tobacco market consists of smuggled cigarettes49 so the true impact of the tax increase may not have been fully considered. We also did not evaluate the rate of tax pass-through for other forms of tobacco such as roll-your-own, or next-generation products as the analysed data sets only contained pricing information for cigarettes, although this is unlikely a serious issue as their use is low in Colombia (eg, heated tobacco product sales made up only 1.2% of Colombia’s tobacco market in 2020).74 75 Similarly, we did not explore all six strategies previously identified as being used by the TI to respond to tax increases (eg, price smoothing), as our data sources did not allow us to do so, so we cannot conclude whether the strategies not examined are present, or not, in the market.

Despite these limitations, this study has several strengths. It is the first academic study we are aware of from Latin America that assess the impact of taxation on the TI pricing strategies, and also the first to examine changes in the price of single cigarettes following a major excise tax increase. Furthermore, the findings of the study augment the evidence base on TI pricing strategies especially in an under-researched context of an LMIC.

### Policy recommendations

Tobacco taxes are still relatively low in Colombia so these should continue to be increased, particularly the specific component so that cheaper economy brands do not have a tax advantage, and hence price differentials between brands narrow. Restrictions on limiting brands to one variant, prohibiting the introduction of new brands/variants, along with a ban on flavours would also likely be effective strategies to adopt.

## Conclusion

In this study we have explored how the industry responds to tax increases in Colombia and have identified the pricing strategies employed by the TI to undermine tax increases. Although Colombia is an LMIC, it still exhibits a lot of the trends we have observed in HICs in terms of the industry tactics. Moreover, we have explored single-stick sales and found that their pricing does not exactly follow the same pattern as packs, suggesting their sales and pricing need to be monitored more carefully. Indeed, it suggests further analysis of single cigarette pricing and its influence on tobacco control policies in LMICs, and thereby on cessation behaviours, is an essential area for future research.

## Data availability statement

The DANE data are freely available from the listed online sources. The DEICS-COL survey data can be shared with researchers who meet the relevant criteria for access—help and information can be obtained directly from Fundación Anáas.

• ## Supplementary Data

This web only file has been produced by the BMJ Publishing Group from an electronic file supplied by the author(s) and has not been edited for content.

## Footnotes

• Twitter @ZainebDanish, @JRBranston, @andinomom, @BathTR

• Contributors All authors contributed to the original idea for the paper. All data were sourced by ZDS, BAL and NM. ZDS designed the study, performed data analysis and interpretation and wrote the first draft of the paper with significant input from JRB, who also provided expertise on taxation and pricing, along with ABG providing critical feedback. All authors contributed to the editing of the final version of the paper. The responsibility for any errors remains entirely with the authors. The corresponding author, JRB, stands as guarantor, accepting full responsiblity for the work and/or the conduct of the study, had access to the data, and controlled the decision to publish.

• Funding ZDS is funded by Cancer Research UK (grant number A29421), and JRB and ABG acknowledge the support of Bloomberg Philanthropies Stopping Tobacco Organizations and Products project funding (www.bloomberg.org).

• Competing interests JRB owns 10 shares in Imperial Brands for research purposes. The shares were a gift from a public health campaigner and are not held for financial gain or benefit. All dividends received are donated to health-related charities, and proceeds from any future share sale or takeover will be similarly donated.

• Provenance and peer review Not commissioned; externally peer reviewed.

• Supplemental material This content has been supplied by the author(s). It has not been vetted by BMJ Publishing Group Limited (BMJ) and may not have been peer-reviewed. Any opinions or recommendations discussed are solely those of the author(s) and are not endorsed by BMJ. BMJ disclaims all liability and responsibility arising from any reliance placed on the content. Where the content includes any translated material, BMJ does not warrant the accuracy and reliability of the translations (including but not limited to local regulations, clinical guidelines, terminology, drug names and drug dosages), and is not responsible for any error and/or omissions arising from translation and adaptation or otherwise.


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# Greatest term!

Algebra Level 3

Consider the binomial expansion $$(2+3x)^9$$. Let $$\mathbb{P}$$ denote the largest coefficient of all terms when it's expanded.

If $$\left ( \dfrac {3}{2} \right )^6 \times \mathbb{P} = \dfrac {3^a \times b}{c}$$

for natural numbers $$a,b,c$$ with $$\text{gcd}(c,3)=\text{gcd}(b,c) = 1$$.

What is the value of $$a+b+c$$?

×

format: plain


text:
# nLab Morava E-theory

Contents

cohomology

### Theorems

#### Stable Homotopy theory

stable homotopy theory

Introduction

# Contents

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

There are several cohomology theories that are being called Morava E-theory at times:

• $B P\langle n\rangle$, the truncated Brown-Peterson spectrum;

• $E(n)$, the Johnson-Wilson spectrum, a localization of $B P \langle n\rangle$ at $v_n$;

• $\widehat{E(n)}$ the complete Johnson-Wilson spectrum

• $E(k,\Gamma)$ the Lubin-Tate spectrum associated to the universal deformation of a formal group law $\Gamma$ over $k$.

## Definition

Choose

• $k$ be a perfect field of characteristic $p$;

• $f$ be a formal group of height $n$ over $k$.

Write

$R \coloneqq W(k)[ [ v_1, \cdots, v_{n-1} ] ]$

for the Lubin-Tate ring of $f$, classifying its universal deformation.

By the discussion there, this is Landweber exact, hence defines a cohomology theory. Therefore by the Landweber exact functor theorem there is an even periodic cohomology theory $E(n)^\bullet$ represented by a spectrum $E(n)$ with the property that its homotopy groups are

$\pi_\bullet(E(n)) \simeq W(k)[ [v_1, \cdots, v_{n-1} ] ] [ \beta^{\pm 1} ]$

for $\beta$ of degree 2. This is called alternatively $n$th Morava E-theory, or Lubin-Tate theory or Johnson-Wilson theory.

(e.g. Lurie, lect 22)

## Properties

### As a localization of the $\infty$-group $\infty$-ring on $B^{n+1}\mathbb{Z}_p$

There is a Snaith theorem for the homotopy fixed points of the Morava E-theory spectrum $E_n$ for the canonical action of a certain group, which identifies these with a localization of the ∞-group ∞-ring on the (n+1)-group $B^{n+1} \mathbb{Z}_p$. (Westerland 12, theorem 1.2)

See at Snaith-like theorem for Morava E-theory for more.

### Bousfield localization and chromatic filtration

The Bousfield localization of spectra $L_{E(n)}$ at $n$th Morava E-theory is called chromatic localization. It behaves on complex oriented cohomology theories like the restriction to the closed substack

$\mathcal{M}_{FG}^{\leq n+1} \hookrightarrow \mathcal{M}_{FG} \times Spec \mathbb{Z}_{(p)}$

of the moduli stack of formal groups on those of height $\geq n+1$.

In this way the localization tower at the Morava E-theories exhibits the chromatic filtration in chromatic homotopy theory.

chromatic homotopy theory

chromatic levelcomplex oriented cohomology theoryE-∞ ring/A-∞ ringreal oriented cohomology theory
0ordinary cohomologyEilenberg-MacLane spectrum $H \mathbb{Z}$HZR-theory
0th Morava K-theory$K(0)$
1complex K-theorycomplex K-theory spectrum $KU$KR-theory
first Morava K-theory$K(1)$
first Morava E-theory$E(1)$
2elliptic cohomologyelliptic spectrum $Ell_E$
second Morava K-theory$K(2)$
second Morava E-theory$E(2)$
algebraic K-theory of KU$K(KU)$
3 …10K3 cohomologyK3 spectrum
$n$$n$th Morava K-theory$K(n)$
$n$th Morava E-theory$E(n)$BPR-theory
$n+1$algebraic K-theory applied to chrom. level $n$$K(E_n)$ (red-shift conjecture)
$\infty$complex cobordism cohomologyMUMR-theory

### Smash product theorem

A version of the smash product theorem

For $X$ a homotopy type/spectrum and for all $n$, there is a homotopy pullback

$\array{ L_{E(n)}X &\longrightarrow& L_{K(n)}X \\ \downarrow && \downarrow \\ L_{E(n-1)}X &\longrightarrow& L_{E(n-1)}L_{K(n)}X } \,,$

where $L_{K(n)}$ denotes the Bousfield localization of spectra at $n$th Morava K-theory and similarly $L_{E(n)}$ denotes localization at Morava E-theory.

### Bousfield equivalence class

For all $n$, $E(n)$ is Bousfield equivalent to $E(n-1) \times K(n)$, where the last factor is $n$th Morava K-theory.

Not to be confused with C*-algebra-E-theory.

## References

Named after Jack Morava (see at Morava K-theory).

Relevant background lecture notes include

and more specifically see the lectures

also

• Report of $E$-theory conjectures seminar (2013) (pdf)

Discussion of the $E_\infty$-algebra structure over $B P$ is in

based on

• Neil Strickland, Products on $MU$-modules, Trans. Amer. Math. Soc. 351 (1999), 2569-2606.

Discussion of twists of Morava E-theory is in

A Snaith theorem-like characterization of Morava E-theory is given in

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text:
# Recovering congruence conditions from the Hilbert class polynomial for idoneal numbers

Before I can ask my question, I need to introduce some terminology and background.

Statement 1: Let $n$ be one of Euler's 65 convenient numbers. Then we can find congruence conditions such that

$$p=x^2+ny^2 \Leftrightarrow c_1, \ldots, c_r \mod{N}$$ See for example this article why this is possible. On a related note: one can even show that for other numbers it is impossible to provide congruence conditions (cf. here, here, and page 19 here)

.

Statement 2: The more general case goes as follows.

.

Statement 3: Finding the HCF for $K=\mathbb{Q}(\sqrt{-n})$ is difficult. However, when $n$ is one of Euler's 65 convenient numbers then the Hilbert class field of $K$ equals the genus field of $K$. The genus field can be easily determined as follows:

Now, basically what I want to do is the following:

• Consider cases when the genus field equals the Hilbert class field (i.e. Euler's convenient numbers)
• Construct a primitive element for the genus field (cf. here)
• Next, from the minimal polynomial of the Hilbert class field, I want to recover congruence conditions through quadratic reciprocity.

Two worked out examples:

1. The genus field of $K=\mathbb{Q}(\sqrt{-5})$ is $M=K(\sqrt{5})=K(i)$. Now, the Hilbert class field equals the genus field, so the minimal polynomial for the primitive element $i$ is $x^2 + 1$. From the above theorem we get the condition that $$x^2 + 1 \equiv 0 \mod{p}$$ must be solvable. But this just means that $-1$ is a quadratic residue modulo $p$. Together with the other condition of the theorem, which says that $-5$ is a quadratic residue modulo $p$, we find that: $$p = x^2 + 5y^2 \Leftrightarrow p \equiv 1,9 \mod{20}$$

2. The genus field of $K=\mathbb{Q}(\sqrt{-21})$ is $M=K(\sqrt{-3},\sqrt{-7})$. To calculate the minimal polynomial for the primitive element $\sqrt{-3} + \sqrt{-7}$, we use Wolfram: $$f(x)=x^4+20x^2+16=(x^2+10)^2-84$$ This polynomial is solvable modulo $p$ if $$y^2 \equiv 84 \mod{p}$$ is solvable, AND $$x^2 \equiv - 10 \mod{p}$$ has a solution. Each condition can, via quadratic reciprocity, be expressed by a congruence, and putting the different congruences together (via the AND operator) gives us again a congruence condition.

Problem:

The above method seems to break down when we consider more than $2$ elements. (eg. $\sqrt{-3}+\sqrt{5}+\sqrt{-7}$). Wolfram does not give a nice "recursive quadratic" polynomial. However, I suspect that congruence conditions can be recovered in a similar way.

-
Why do you want to use this roundabout method? Finding the primes represented by all the forms in a genus, taken together, is pretty easy, see Cox Corollary 2.27, page 36 for principal genus. If there is only one form (class) per genus, finished. –  Will Jagy Apr 17 '14 at 18:21
For example, $x^2 + 105 y^2$ represents all primes $p$ with $$p \equiv 1, 109, 121, 169, 289, 361 \pmod {420}.$$ Note that $23^2 - 420 = 109.$ –  Will Jagy Apr 17 '14 at 18:26
I am aware of the approach via genus theory, quadratic forms, etc. The motivation of my question is the following: I am writing my MA thesis and can 'only' spend 40-50 pages. I decided to leave out PART I of Cox. However, I hoped that I could recover some important results of it with the techniques of PART II. –  yannickvda Apr 17 '14 at 18:28
Take a look at Hudson_Williams and Liu_Williams at zakuski.utsa.edu/~jagy/inhom.html , you might as well know what really happens with more than one per genus. –  Will Jagy Apr 17 '14 at 18:33


text: If the isotope that you wish to decay is not on the drop down list, check the 'not listed' check-box and manually enter the isotope name and its half-life to perform the calculation. Incidentally, our formula for t gives us an easy way of finding the half-life, the time it takes for half the nuclei in a sample to decay.The half-life (often denoted t 1/2) is just t(1/2) = (1/l) ln(2).The equivalent thickness for the medium in … (The rate of decay of undecayed nuclei N is proportional to the number N of undecayed nuclei present.) In this first chart, we have a radioactive substance with a half life of 5 years. Formula ; Online calculator that allows you to find out the radiation activity decay in most popular isotopes used in chemistry and medicine. Kindly read the relevant sections in Krane’s book ﬁrst. Exponential growth and decay show up in a host of natural applications. Code to add this calci to your website . Anybody can ask a question Anybody can answer The best answers are voted up and rise to the top Home Questions Tags Users Unanswered Radioactive Decay Function. It only takes a minute to sign up. which is a very tidy result! Decay Formula Exponential problems usually move around the decay formula in mathematics. Divide by 3.2 on both sides. The equation for radioactive decay is: where is the inital number of nuclei is the number of nuclei after a specific time period is the decay constant (a higher decay constant means a faster radioactive decay) is the time period. Note: The calculation of radioactivity in minerals is based on certain assumptions. As you can see, the substance initially has 100% of its atoms, but after its first half life (5 years) only 50% of the radioactive atoms are left. Growth and Decay If a quantity y is a function of time t and is directly proportional to its rate of change (y’) , then we can express the simplest differential equation of growth or decay. b) how long would it take this sample to decay to 20% of its original amount. Half Life Calculus? One of the most prevalent applications of exponential functions involves growth and decay models. Exponential growth and decay. You want to know the decay rate of element C. Unfortunately, you can’t find the decay chart that you normally use. Since we start with 800 mg, then we know that 800) 0 P0 = . Half-life (symbol t 1⁄2) is the time required for a quantity to reduce to half of its initial value.The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo, or how long stable atoms survive, radioactive decay.The term is also used more generally to characterize any type of exponential or non-exponential decay. The radioactive decay process occurs when some original or parent nucleus of an unstable atom decomposes and it forms a different nucleus or we can call it the daughter nucleus too. The most famous example is radioactive decay. This reading is supplementary to that, and the subsection ordering will mirror that of Krane’s, at least until further notice. Indeed, radioactive material follows an exponential decay equation, and each material has (depending on its own volatility) its half-time, which is the amount of time it takes for the amount of radioactive material to reduce to half. Many systems exhibit exponential growth. One of which is growth and decay – a simple type of DE application yet is very useful in modelling exponential events like radioactive decay, and population growth. Usually, the formula for radioactive decay is … It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. The pressure at sea level is about 1013 hPa (depending on weather). Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Exponential decay occurs in a wide variety of cases that mostly fall into the domain of the natural sciences. Take the natural log of both sides to eliminate the exponential. A simplified radioactive decay equation has been obtained by combining the principles of sequences and series with the radioactive decay equation. Episode 515: The radioactive decay formula. It is noted that the radio-isotopes do not have enough binding energy to hold the nucleus in the atom. Step 1. Sign up to join this community. The radioactive decay law states that the probability per unit time that a nucleus will decay is a constant, independent of time. In calculus notation, this is. Radioactive Decay Formula is provided here by our subject experts. This is the underlying relationship in any process that follows exponential decay. The above is a substitute static image See About the calculus applets for operating instructions. Using the exponential decay formula to calculate k, calculating the mass of carbon-14 remaining after a given time, and calculating the time it takes to have a specific mass remaining . In the lab, there's another radioactive element, C, which has decayed to half of its mass in 24 hours. The radioactive half-life for a given radioisotope is a measure of the tendency of the nucleus to "decay" or "disintegrate" and as such is based purely upon that probability. The tiny nuclear size compared to the atom and the enormity of the forces which act within it make it almost totally impervious to the outside world. The rate at which radioactive decay process happens is measured with the help of half-life that is defined as the total time for the amount of parent nucleus to decay. If you're seeing this message, it means we're having trouble loading external resources on our website. According to this model the mass $$Q(t)$$ of a radioactive material present at time $$t$$ satisfies Equation \ref{eq:3.10.1}, where $$a$$ is a negative constant whose value for any given material must be determined by experimental observation. Introduction to Exponential Decay. Download Radioactive Decay Formula along with the complete list of important formulas used in maths, physics & chemistry. In Physics, the radioactive decay is defined as the spontaneous breakdown of the nucleus, which results in the release of energy and also the matter from the atomic nucleus. Write the formula (with its "k" value), Find the pressure on the roof of the Empire State Building (381 m), and at the top of Mount Everest (8848 m) Start with the formula… The decay formula […] It decreases about 12% for every 1000 m: an exponential decay. You may also back decay sources to find out the original activity (or for any date), knowing the current activity. We have seen that the solution for dy/dx = y is an exponential function. d N d t = − λ × N. Ask your students to put this equation into words. ... Have the students use the modified radiation decay equation to determine how long it will take for each of the following radionuclides to decay “completely” if the total amount were deposited in one place. Radioactive Decay. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. Divide by negative two on both sides. Explain the concept of half-life. The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. To show money, bacteria, fishes in a pond, the exponential growth or decay formula is used frequently. Where continuous growth or decay are shown in the form of small r and t is the time during which decay was measured. The exponential decay is found in processes where amount of something decreases at a rate proportional to its current value. Decay Law – Equation – Formula. Radioactive Decay Note to students and other readers: This Chapter is intended to supplement Chapter 6 of Krane’s excellent book, ”Introductory Nuclear Physics”. Ask Question Asked 5 years, 2 … Book: Calculus (OpenStax) ... Use the exponential decay model in applications, including radioactive decay and Newton’s law of cooling. Use the given half-life to find the decay rate r by setting up an equation and choosing P and P₀ such that the final amount (P) is equal to half the starting amount (P₀). This chemistry video tutorial shows explains how to solve common half life radioactive decay problems. Convert this to a percentage. Radioactive decay? Write your decay formula: y = Ae-kt It is usually used to describe quantities undergoing exponential decay (for example, radioactive decay) where the half-life is constant over the whole life of the decay, and is a characteristic unit (a natural unit of scale) for the exponential decay equation. Radioactive Decay Useful for calculating today's activity for any radioactive isotope. Write the formula for radioactive decay. Sample Problem: After 500 years, a sample of radium-226 has decayed to 80.4% of its original mass. Radioactive decay is a first order rate reaction, so the expression for the rate is: log 10 X 0 /X = kt/2.30 where X 0 is the quantity of radioactive substance at zero time (when the counting process starts) and X is the quantity remaining after time t . This device cannot display Java animations. The formula the the amount P remaining after exponential growth/decay is: P = P₀e^(rt) Where r is the decay rate, t is time, and P₀ is the initial amount. Experimental evidence shows that radioactive material decays at a rate proportional to the mass of the material present. Half-life is the period of time it takes for a substance undergoing decay to decrease by half. In this section, we examine exponential growth and decay in the context of some of these applications. Growth, Decay, and the Logistic Equation. Math Standards Measurement • Use unit analysis to check measurement computations. This constant probability may vary greatly between different types of nuclei, leading to the many different observed decay rates. On this page we explore this a bit more. This element's decay rate is approximately: Solution: (Part a) Since this is an exponential decay problem, we will use the formula P t =P e−kt ( . From population growth and continuously … Exponential Growth Model. top; Formula; Real World Ex; Graphs ; Radioactive Decay Overview. Substitute the values in the equation and solve for lambda. 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text:
# Nine-gon

Calculate the perimeter of a regular nonagon (9-gon) inscribed in a circle with a radius 13 cm.

Result

x =  80.03 cm

#### Solution:

$\theta = \dfrac{ 180 ^\circ }{ 9} = \dfrac{ \pi}{ 9} = 20 ^\circ \ \\ a = 2r \sin( \dfrac{ \pi}{ 9} ) = 8.89 \ cm \ \\ x = 9a = 80.03 \ \text{cm}$

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Peter Amrehn
@jongleur1983
is it known/as designed, that the Pipeline Tool (Gui) crashes whenever a referenced dll with importers is rebuilt (without the pipeline currently building itself)?
Nikos Kastellanos
@nkast
in the context object you will find a .Logger . Use this to log messages and warnings.
for errors throw an InvalidContentException or a PipelineException. That's the recommended way.
if you throw another type of exception it gets wrapped and reported as 'Unexpected Error in Importer/Processsor' and the original error message gets lost/obscured.
Nikos Kastellanos
@nkast
@jongleur1983 The pipeline loads the ref .dlls and scan them for importers/exporters , as the result the DLLs are locked by the system.
is it possible to debug your importer by invoking the MGCB directly? say for example set MGCB as 'start external program' in debug properties with the .mgcb file as a parameter...
Peter Amrehn
@jongleur1983
@nkast thanks. found the logger, and that's fine. I just wished I could log something that's shown inside the green success messages as well when output is filtered. I know how debugging works now though ;)
Peter Amrehn
@jongleur1983
is there a good way to learn shader programming? My goal is to migrate some old XNA3.1 shaders for MonoGame. I don't know what the old ones do exactly, I don't know the shader language, but I'd like to keep functionality as close as possible to the original one. Is there some kind of reference on how a shader looks like and works (ideally for the old XNA shaders AND for current monogame pipeline)?
Simon (Darkside) Jackson
@SimonDarksideJ
The biggest issue with updating old 3.1 shaders is keeping backwards compatibility. Easiest wat is to have an #IF on the shader version support and flipping to SM4.
However, it's not a magic bullet and sometimes doesn't work with features that are deprecated sadly.
Daniel Sör
@Ragath
if they're on 3.1 then i think we can assume they're in a read-only state by now, so a 1-way conversion should be fine
Peter Amrehn
@jongleur1983
@SimonDarksideJ what backwards compatibility are you talking about? I don't need to be compatible to the original platforms or similar. I try to re-implement (as just migrating is a worthless-to-go rabbit hole) the game on the current framework, just to get it running again.
will-scc
@will-scc
Hi guys, if I wanted to use the latest development build in an existing project, just running the installer from http://teamcity.monogame.net/repository/download/MonoGame_PackagingWindows/latest.lastSuccessful/MonoGameSetup.exe?guest=1 should be sufficient, right?
YTN0
@YTN0
Is the dev nugets functional for 3.8? I tried converting one of my projects over from the installer version to the 3.8 nuget version and I couldn't get my content pipeline stuff to compile. It kept complaining about assimp64.dll when trying to compile an fbx
Harry
@harry-cpp
They are functional
Peter Amrehn
@jongleur1983
Hi. I try to get the simplest possible version of a shader to work but keep getting "error X3003: redefinition of VertexShaderOuptut" for a struct I define, following this tutorial: https://www.reddit.com/r/gamedev/comments/435dkp/how_to_add_pixel_shaders_to_monogame/ (without calculating anything, skipping World and View and just returning input except normal)
Daniel Sör
@Ragath
@jongleur1983 you need the shader to actualy do stuff, or the compiler is going to strip everything out in the optimizing pass
Peter Amrehn
@jongleur1983
@Ragath thanks, that did the trick...
Peter Amrehn
@jongleur1983
compiling fine with the content pipeline, but failing to load now :/
Peter Amrehn
@jongleur1983
YTN0
@YTN0
@harry-cpp are there instructions somewhere on setting up and using the nugets instead of the installer? I've not had any luck getting my pipeline content to compile. Is there a specific version of .net core or .net standard that we should be targeting? Also, where do the nuget templates come up? I also could not get the monogame core nuget to install (noth the DX or GL version). It also seems out of date.
YTN0
@YTN0

The specific issues are I can't compile any pipeline content that require Assimp. The error I get is:

2>SkyBoxCube.fbx : error : Importer 'FbxImporter' had unexpected failure!
2>Assimp.AssimpException: Error loading unmanaged library from path: Assimp64.dll, see inner exception for details.
2>The specified module could not be found. (0x8007007E)
2> ---> System.IO.FileNotFoundException: The specified module could not be found. (0x8007007E)

Separately, I can't seem to compile any shaders that reference or include other files e.g.

2>EXEC : error : C:\Users\default\AppData\Local\Temp\tmp8C31.tmp(1,0): File not found: Lighting.fx in .(MonoGame.Effect.Preprocessor+MGFile) C:\Users\default\AppData\Local\Temp
2>Failed to parse 'C:\Users\default\AppData\Local\Temp\tmp8C31.tmp'!

Jay Mann
@jaym2444
When trying to initialize my own graphics device so I can interop it within a UI Library, I always get thrown an error about a null pointer in the default graphics adapter
Harry
@harry-cpp
@YTN0 the second issue has been fixed with the newest develop version
report it on github
YTN0
@YTN0
@harry-cpp Thanks for the update. Will file an issue for the 1st problem. For the second issue, has that just been resolved? I am currently using the latest build, which was from yesterday (5/23)
Harry
@harry-cpp
Resolved 4 days ago: MonoGame/MonoGame#7148
Peter Amrehn
@jongleur1983
Hi. Is it possible to have dependencies between Assets in the content pipeline (it was possible on Xna3.1)? My ContentProcessor writes ExternalReference<TextureContent> to the tag of ModelContent objects, but the contentPipeline fails to compile the model as "TextureImporter failed to load", on the other hand, the same texture as it's own asset compiles fine.
marinariverside
game này có gì hấp dẫn
Peter Amrehn
@jongleur1983
I tried to upgrade my MonoGame packages to MonoGame.Framework.WindowsDX 3.8.0 prerelease - now some real basic types can't be found any more: Vector2, Vector3, Matrix, ContentTypeReader<T>, ContentReader - what am I doing wrong?
Jesse
@Jjagg
@jongleur1983 What .NET do you target? Try deleting bin and obj before building.
Peter Amrehn
@jongleur1983
@Jjagg trying to target netcoreapp3.0
netcoreapp3.1 would be ideal though
Peter Amrehn
@jongleur1983
@Jjagg Monogame.Framework.Content.Pipeline.Portable can't be restored using .netcoreapp3.1 - is there a proper replacement for that?
wait... I might have found the remaining issue, I'll try further...
Peter Amrehn
@jongleur1983
no... don't get it right, yet...
Let me ask the other way around: What to use under the following conditions: I need to use a library that targets netcoreapp3.1 and netstandard2.1; I need a content project (preferably but not necessarily using the Pipeline UI tool) with custom ContentProcessors and ContentImporters.
oh: and using DirectX under Windows for now.
Kevin Small
@KevinSmall
I have a problem building an iOS game from VS in Windows.
I have a MG 3.7.1 game developed in VS for Windows targeting iOS. I use the VS "Pair to Mac" feature to get the iPhone simulator running on Windows. My Mac is a mac-in-cloud Mac, not sure if that matters.
When I choose soln config = Debug, platform = iPhoneSimulator, I can build and run game on iPhoneSimilator in Windows. All good, all works.
When I choose soln config = Release, platform = iPhone I get an error at build time, saying "Failed to resolve assembly: 'Lidgren.Network". If I unpair my Mac it builds fine, so I suspect something is missing on the Mac. The Mac has VS and MG 3.7.1 installed.
Can anyone help?
Kevin Small
@KevinSmall
plus one slightly easier question, does anyone know what happened to Guide.cs that used to exist in 3.5.1, it seems to be gone in latest MG release.
Kevin Small
@KevinSmall
(please ignore my original question about lidgren.network - I worked around it by adding monogame.iOS project directly to my solution)
Jesse
@Jjagg
Guide has been removed. If you used it to show the keyboard check out KeyboardInput
Peter Amrehn
@jongleur1983
I installed the up to date Pipeline tool as a dotnet global too. It loads the content project (without any referenced dll). Whenever I try to add a reference to my compile pipeline project (containing the importers) it crashes, showing the following stack trace on restart: https://gist.github.com/jongleur1983/dff921fe31b44f8a9025a4a265ef7839
Peter Amrehn
@jongleur1983
fixed that issue myself: somehow the colors in my content project had the wrong format. Up to the next: content pipeline compiles all assets successfully (it says), but the FbxImporter fails to load fbx assets with "Assimp.AssimpException: Error loading unmanaged library from path: Assimp64.dll". What am I missing here?
Kevin Small
@KevinSmall
Thanks @Jjagg
Dan Gant
@dgant

Hello. Using MonoGame for the first time. Created a new project from the MonoGame cross-platform project template in Visual Studio. But the project template has NuGet package errors out of the box:

• Unable to find package MonoGame.Content.Builder with version (>= 3.8.0) | Found 9 version(s) in nuget.org [ Nearest version: 3.7.0.9 ]
• Unable to find a stable package MonoGame.Framework.DesktopGL with version (>= 3.8.0) | Found 7 version(s) in nuget.org [ Nearest version: 3.8.0.1375-develop ]

What's strange is that it looks like I have 3.8 installed (see screenshot).

What would get the packages lined up in a working state?


text:
## 4.5 Acceleration Methods

Dempster et al. showed that the convergence rate for the EM algorithm is linear, which can be painfully slow for some problems. Therefore, a cottage industry has developed around the notion of speeding up the convergence of the algorithm. Two approaches that we describe here are one proposed by Tom Louis based on the Aitken acceleration technique and the SQUAREM approach of Varadhan and Roland.

### 4.5.1 Louis’s Acceleration

If we let $$M(\theta)$$ be the map representing a single iteration of the EM algorithm, so that $$\theta_{n+1} = M(\theta_n)$$. Then under standard regularity conditions, we can approximate $$M$$ near the optimum value $$\theta^\star$$ with

$\theta_{n+1} = M(\theta_{n}) \approx \theta_n + J(\theta_{n-1})(\theta_n - \theta_{n-1})$

where Dempster et al. 1977 showed that $$J$$ is

$J(\theta) = I_{Z|Y}(\theta)I_{Z,Y}(\theta)^{-1},$

which can be interpreted as characterizing the proportion of missing data. (Dempster et al. also showed that the rate of convergence of the EM algorithm is determined by the modulus of the largest eigenvalue of $$J(\theta^\star)$$.)

Furthermore, for large $$j$$ and $$n$$, we have

$\theta_{n + j + 1} - \theta_{n+j} \approx J^{(n)}(\theta^\star)(\theta_{j+1}-\theta_{j})$

where $$\theta^\star$$ is the MLE, and $$J^{(n)}(\theta^\star)$$ is $$J$$ multiplied by itself $$n$$ times. Then if $$\theta^\star$$ is the limit of the sequence $$\{\theta_n\}$$, we can write (trivially) for any $$j$$ $\theta^\star = \theta_j + \sum_{k=1}^\infty (\theta_{k + j} - \theta_{k+j-1})$

We can then approximate this with

$\begin{eqnarray*} \theta^\star & \approx & \theta_j + \left( \sum_{k = 0}^\infty J^{(k)}(\theta^\star) \right) (\theta_{j+1}-\theta_j)\\ & = & \theta_j + (I-J(\theta^\star))^{-1} (\theta_{j+1}-\theta_j) \end{eqnarray*}$

The last equivalence is possible because the eigenvalues of $$J$$ are all less than one in absolute value.

Given this relation, the acceleration method proposed by Louis works as follows. Given $$\theta_n$$, the current estimate of $$\theta$$,

1. Compute $$\theta_{n+1}$$ using the standard EM algorithm

2. Compute $$(I-\hat{J})^{-1} = I_{Y,Z}(\theta_n)I_Y(\theta_n)^{-1}$$

3. Let $$\theta^\star = \theta_n+(I-\hat{J})^{-1}(\theta_{n+1}-\theta_n)$$.

4. Set $$\theta_{n+1} = \theta^\star$$.

The cost of using Louis’s technique is minimal if the dimension of $$\theta$$ is small. Ultimately, it comes down to the cost of inverting $$I_Y(\theta_n)$$ relative to running a single iteration of the EM algorithm. Further, it’s worth emphasizing that the convergence of the approach is only guaranteed for values of $$\theta$$ in a neighborhood of the optimum $$\theta^\star$$, but the size and nature of that neighborhood is typically unknown in applications.

Looking at the algorithm described above, we can gather some basic heuristics of how it works. When the information in the observed data is high relative to the complete data, then the value of $$(I-\hat{J})^{-1}$$ will be close to $$1$$ and the sequence of iterates generated by the algorithm will be very similar to the usual EM sequence. However, if the proportion of missing data is high, then $$(I-\hat{J})^{-1}$$ will be much greater than $$1$$ and the modifications that the algorithm makes to the usual EM sequence will be large.

#### 4.5.1.1 Example: Normal Mixture Model

Recall that the data are generated as follows.

mu1 <- 1
s1 <- 2
mu2 <- 4
s2 <- 1
lambda0 <- 0.4
n <- 100
set.seed(2017-09-12)
z <- rbinom(n, 1, lambda0)
y <- rnorm(n, mu1 * z + mu2 * (1-z), s1 * z + (1-z) * s2)
hist(y)
rug(y)

If we assume $$\mu_1$$, $$\mu_2$$, $$\sigma_1$$ and $$\sigma_2$$ are known, then we can visualize the observed data log-likelihood as a function of $$\lambda$$.

f <- function(y, lambda) {
lambda * dnorm(y, mu1, s1) + (1-lambda) * dnorm(y, mu2, s2)
}
loglike <- Vectorize(
function(lambda) {
sum(log(f(y, lambda)))
}
)
curve(loglike, 0.01, 0.95, n = 200, xlab = expression(lambda))

Because the observed log-likelihood is relatively simple in this case, we can maximize it directly and obtain the true maximum likelihood estimate.

op <- optimize(loglike, c(0.01, 0.95), maximum = TRUE, tol = 1e-8)
op$maximum [1] 0.3097386 We can encode the usual EM iteration as follows. The M function represents a single iteration of the EM algorithm as a function of the current value of $$\lambda$$. make_pi <- function(lambda, y, mu1, mu2, s1, s2) { lambda * dnorm(y, mu1, s1) / (lambda * dnorm(y, mu1, s1) + (1 - lambda) * (dnorm(y, mu2, s2))) } M <- function(lambda0) { pi.est <- make_pi(lambda0, y, mu1, mu2, s1, s2) mean(pi.est) } We can also encode the accelerated version here with the function Mstar. The functions Iy and Iyz encode the observed and complete data information matrices. Iy <- local({ d <- deriv3(~ log(lambda * dnorm(y, mu1, s1) + (1-lambda) * dnorm(y, mu2, s2)), "lambda", function.arg = TRUE) function(lambda) { H <- attr(d(lambda), "hessian") sum(H) } }) Iyz <- local({ d <- deriv3(~ pihat * log(lambda) + (1-pihat) * log(1-lambda), "lambda", function.arg = TRUE) function(lambda) { H <- attr(d(lambda), "hessian") sum(H) } }) Mstar <- function(lambda0) { lambda1 <- M(lambda0) pihat <- make_pi(lambda0, y, mu1, mu2, s1, s2) lambda0 + (Iyz(lambda0) / Iy(lambda0)) * (lambda1 - lambda0) } Taking a starting value of $$\lambda = 0.1$$, we can see the speed at which the original EM algorithm and the accelerated versions converge toward the MLE. lambda0 <- 0.1 lambda0star <- 0.1 iter <- 6 EM <- numeric(iter) Accel <- numeric(iter) for(i in 1:iter) { pihat <- make_pi(lambda0, y, mu1, mu2, s1, s2) lambda1 <- M(lambda0) lambda1star <- Mstar(lambda0star) EM[i] <- lambda1 Accel[i] <- lambda1star lambda0 <- lambda1 lambda0star <- lambda1star } results <- data.frame(EM = EM, Accel = Accel, errorEM = abs(EM - op$maximum),
errorAccel = abs(Accel - op\$maximum))

After six iterations, we have the following.

format(results, scientific = FALSE)
         EM     Accel      errorEM        errorAccel
1 0.2354541 0.2703539 0.0742845130 0.039384732683244
2 0.2850198 0.3075326 0.0247188026 0.002206035238572
3 0.3014516 0.3097049 0.0082869721 0.000033703087859
4 0.3069478 0.3097384 0.0027907907 0.000000173380975
5 0.3087971 0.3097386 0.0009414640 0.000000006066448
6 0.3094208 0.3097386 0.0003177924 0.000000005785778

One can see from the errorAccel column that the accelerated method’s error decreases much faster than the standard method’s error (in the errorEM column). The accelerated method appears to be close to the MLE by the third iteration whereas the standard EM algorithm hasn’t quite gotten there by six iterations.

### 4.5.2 SQUAREM

Let $$M(\theta)$$ be the map representing a single iteration of the EM algorithm so that $$\theta_{n+1} = M(\theta_n)$$. Given the current value $$\theta_0$$,

1. Let $$\theta_1 = M(\theta_0)$$

2. Let $$\theta_2 = M(\theta_1)$$

3. Compute the difference $$r = \theta_1-\theta_0$$

4. Let $$v = (\theta_2-\theta_1) - r$$

5. Compute the step length $$\alpha$$

6. Modify $$\alpha$$ if necessary

7. Let $$\theta^\prime = \theta_0 - 2\alpha r + \alpha^2 v$$

8. Let $$\theta_1 = M(\theta^\prime)$$

9. Compare $$\theta_1$$ with $$\theta_0$$ and check for convergence. If we have not yet converged, let $$\theta_0 = \theta_1$$ and go back to Step 1.


text:
# Module 20 Chi-Square Test

Situations where a categorical response variable is recorded would be summarized with a frequency or percentage table (see Modules 4 and 10). The appropriate test statistic in these situations is a chi-square rather than a t. The Chi-Square Test test statistic follows a chi-square distribution, which is introduced below. The rest of this module is dedicated to the general Chi-Square Test where the distribution of a categorical response variable is compared between two or more groups (or populations). The related goodness-of-fit test for a categorical response recorded for only one group (or population) is introduced in Module 21.

## 20.1 Chi-Square Distribution

A chi-square (χ2) distribution is generally right-skewed (Figure 20.1), with the exact shape dictated by the degrees-of-freedom (df); as the df increase the sharpness of the skew decreases (Figure 20.1).

In its simplest form, the χ2 distribution arises as a sampling distribution for the χ2 test statistic,

$\chi^{2} = \sum_{cells}\frac{(Observed-Expected)^{2}}{Expected}$

where “Observed” and “Expected” represent the observed and expected counts in the cells of frequency tables (see Module 4 and Module 10). Thus, the χ2 distribution arises from comparing frequencies in two tables.71

Unlike the normal and t distributions, the χ2 distribution always represents the two-tailed situation, although the “two tails” will appear as one tail on the right side of the distribution. The simplest explanation for this characteristic is that the “squaring” in the calculation of the χ2 test statistic results in what would be a “negative tail” being “folded over” onto what is the “positive tail.” Thus, all probability (i.e., area) calculations on a χ2 distribution represent the two-tailed alternative hypotheses.

Proportional areas on a χ2 distribution are computed with disrib(). The major differences with a χ2 distribution is that distrib="chisq" must be used in distrib() and the degrees-of-freedom must be given to df=.72 In addition, lower.tail=FALSE is always used when computing a p-value because the upper-tail probability represents the two-tailed alternative hypothesis inherent to all Chi-Square Tests. For example, the area right of χ2=6.456 on a χ2 distribution with 2 df is 0.0396 (Figure 20.2).

( distrib(6.456,distrib="chisq",df=2,lower.tail=FALSE) )

## 20.2 Chi-Square Test Specifics

Researchers commonly want to compare the distribution of individuals into the levels of a categorical variable among two or more groups (or populations). For example, researchers may want to determine if the distribution of failing students differs between males and females, if the distribution of kids playing sports differs between kids from high- or low-income families, if the distribution of four major plant species differs between two locations, or if the distribution of responses to a five-choice question differs between respondents from neighboring counties. All of these questions have a categorical response variable (fail or not, play sport or not, plant species, answer to five-choice question) compared among two or more groups (gender, income category, two locations, neighboring counties). The Chi-Square Test, the subject of this module, can be used for each of these situations.73

### 20.2.1 Hypotheses

The statistical hypotheses for a Chi-Square Test are “wordy.” To explore this, let’s first assume that a two-way frequency table (see Module 10) will summarize the data where the rows correspond to separate groups and the columns correspond to levels of the response variable. In this organization, the Chi-Square Test null hypothesis is that the row percentages are equal – i.e., “the percentage distribution of individuals into the levels of the response variable is the same for all groups.” The alternative hypothesis states that there is some difference among the row percentages – i.e., “the percentage distribution of individuals into the levels of the response variable is NOT the same for all groups.”

As one example (more are shown below), consider the following:

An association of Christmas tree growers in Indiana sponsored a survey of Indiana households to help improve the marketing of Christmas trees. Of the 261 rural households, 64 had a natural tree (as compared to an artificial tree). Of the 160 urban households, 89 had a natural tree. Use these results to determine, at the 10% level, if the distribution of households with a natural tree differed between rural and urban households.

The hypotheses for this situation are,

• H0: “the distrubution of households into the tree types is the same for urban and rural households”
• HA: “the distrubution of households into the tree types is NOT the same for urban and rural households”

### 20.2.2 Tables

As noted above, all two-way frequency tables used for a Chi-Square Test will be organized such that the response variable forms the columns and the groups to be compared form the rows. With this organization, the row-percentage table becomes the table of primary interest because it relates directly to the hypotheses described above. The question of a Chi-Square Test then becomes one of determining whether each row of the row-percentage table is equal, given sampling variability.

The observed raw data must be organized into a two-way frequency table as described in Module 10. For example, the Christmas tree data is summarized as in Table 20.1. The actual calculations for a Chi-Square Test are performed on this observed table. However, the hypothesis test, as described above, is best viewed as a method to determine if each row of the row-percentage table is statistically equivalent or not. Thus, the row-percentage table computed from the frequency table is useful when interpreting the results of a Chi-Square Test (Table 20.2).

Table 20.1: Frequency of individuals in urban and rural households that have a natural or an artificial Christmas tree.
Natural Artificial Sum
Urban 89 172 261
Rural 64 96 160
Sum 153 268 421

Table 20.2: Percentage of individuals within urban and rural households that have a natural or an artificial Christmas tree.
Natural Artificial Sum
Urban 34.1 65.9 100
Rural 40.0 60.0 100
ALL 36.3 63.7 100

The Chi-Square Test requires constructing a table of expected values that are derived from the null hypothesis. Specifically, the “expected” table contains the expected frequency of individuals in each level of the response variable for each group assuming that the distribution of responses does not differ among groups. These expected tables are computed from the margins of the observed table, but are best explained with an illustrative example.

In the Christmas tree example, the null hypothesis states that there is no difference in the distribution of households with a natural tree between the rural and urban areas. Thus, under this null hypothesis, one would expect the proportion (or percentage) of households with a natural tree to be the same in both groups. The proportion of households with a natural tree, regardless of location, is $$\frac{153}{421}$$=0.363. Thus, under the null hypothesis, the proportion of rural AND the proportion of urban households with a natural tree is 0.363. Because there is a different number of urban and rural households in the study, the actual NUMBER (rather than proportion) of households expected to have a natural tree will differ. The NUMBER of urban households expected to HAVE a natural tree is found by multiplying the number of urban households by the common proportion computed above – i.e., 261×0.363=94.743. The remaining urban households would be expected to NOT have a natural tree – i.e., 261-94.743=261×(1-0.363)=166.257. Similar calculations are made for the rural households (i.e., 160×0.363=58.080 expected to have a natural tree and 160×(1-0.363)=101.920 expected to NOT have a natural tree.

These expected frequencies are computed directly and easily from the marginal totals of the observed frequency table (Table 20.1). For example, substituting the fractional representation of the decimal proportions into the calculation of the expected number of urban households with a natural tree gives 261×$$\frac{153}{421}$$=$$\frac{261\times153}{421}$$=94.85374. A close examination of this formula and the marginal totals in Table 20.1 shows that this value is equal to the product of the corresponding row and column marginal totals in the observed table divided by the total number of individuals. The other expected values follow a similar pattern as follows,

• 261×$$\frac{268}{421}=\frac{261\times268}{421}$$=166.147 urban households to NOT have a natural tree.
• 160×$$\frac{153}{421}=\frac{160\times153}{421}$$=58.147 rural households to have a natural tree.
• 160×$$\frac{268}{421}=\frac{160\times268}{421}$$=101.853 rural households to NOT have a natural tree.

Thus, all expected values in a Chi-Square Test are calculated by multiplying the row and column totals of the frequency table and dividing by the total number of individuals. These expected values are summarized in a two-way table, called the expected frequencies table (Table 20.3)

Table 20.3: Expected frequency of individuals in urban and rural households that have a natural or an artificial Christmas tree.
Natural Artificial Sum
Urban 94.853 166.147 261
Rural 58.147 101.853 160
Sum 153.000 268.000 421

### 20.2.3 Specifics

The Chi-Square Test is characterized by a categorical response variable recorded for two or more groups (or populations). The specifics of the Chi-Square Test are below.

• Null Hypothesis: “The distribution of individuals into the levels of the response variable is the same for all groups”
• Alternative Hypothesis: “The distribution of individuals into the levels of the response variable is NOT the same for all groups.”
• Statistic: Observed frequency table.
• Test Statistic: χ2=$$\sum_{\text{cells}}\frac{(\text{Observed}-\text{Expected})^{2}}{\text{Expected}}$$
• df: (r-1)×(c-1) where r is number of rows and c is the number of columns
• Assumptions: Expected value for each category is ≥5.
• Use with: Categorical response, two or more groups (or populations).

A confidence region is not usually constructed for a Chi-Square Test because of the complexity of the statistics and parameter. Thus, Step 11 for a hypothesis test will not be computed for a Chi-Square Test.

## 20.3 Examples

### 20.3.1 Christmas Trees

Below are the 11-steps (Section 17.1) for a full hypothesis test for the Christmas tree example introduced above.

1. α=0.10.
2. H0: “distribution of households by type of tree is the same for urban and rural households” vs. HA: “distribution of households by type of tree is NOT the same for urban and rural households.”
3. A Chi-Square Test is required because …
1. a categorical response variable was recorded (type of tree) and
2. two groups are being compared (urban and rural households).
4. The data appear to be part of an observational study with no clear indication of randomization.
5. The expected frequency in each of the four cells is greater than five (Table 20.3).
6. The observed frequency table is in Table 20.1.
7. χ2 = $$\frac{(89-94.853)^{2}}{94.853}$$ + $$\frac{(172-166.147)^{2}}{166.147}$$ + $$\frac{(64-58.147)^{2}}{58.147}$$ + $$\frac{(96-101.853)^{2}}{101.853}$$ = 0.3611+0.2062+0.5891+0.3363 = 1.4927 with 1 df.
8. p-value=0.2218.
9. H0 is not rejected because the p-value is >α.
10. There does not appear to be a significant difference in the distribution of Christmas tree types among rural and urban households.
11. Not performed for Chi-Square Test.

#### R Appendix:

( distrib(1.4927,distrib="chisq",df=1,lower.tail=FALSE) )

### 20.3.2 Dolphin Behavior

Researchers observed groups of dolphins off the coast of Iceland near Keflavik in 1998. The researchers recorded the time of the day and the main activity of the group, whether traveling quickly, feeding, or socializing. The number of dolphin groups observed by each time of day and activity is shown in Table 20.4. Use this information to determine, at the 5% level if the proportion of groups exhibiting each activity differs by time of day.

Table 20.4: Observed frequencies of dolphin observations in each activity type and time of day.
Travel Feed Social Sum
Morning 6 28 38 72
Noon 10 7 9 26
Afternoon 15 1 10 26
Evening 13 56 10 79
Sum 44 92 67 203

1. α = 0.05.
2. The H0: “The distribution of dolphin groups into each activity type is the same among times of day” versus HA: “The disribution of dolphin groups into each activity type is NOT the same among times of day.”
3. A chi-square test is required because
1. a categorical variable with three levels (activity type) was measured and
2. four groups (times of day) are being compared.
4. An observational study that is not obviously a random sample was used.
5. The test statistic below will follow a χ2 distribution because the expected number in each cell is greater than five (Table 20.5).
Table 20.5: Expected frequencies of dolphin observations in each activity type and time of day.
Travel Feed Social Sum
Morning 15.6 32.6 23.8 72
Noon 5.6 11.8 8.6 26
Afternoon 5.6 11.8 8.6 26
Evening 17.1 35.8 26.1 79
Sum 44.0 92.0 67.0 203

1. The table of observed values is in Table 20.4.
2. The χ2 test statistic is 68.400 (=5.913 + 3.380 + 15.561 + 0.993 + 0.657 + 1.942 + 9.868 + 11.393 + 8.529 + 0.020 + 0.235 + 9.909) with 6 df (=(4-1)×(3-1)).
3. The p-value for this test statistic is <0.00005 (more specifically it is 8.7e-13).
4. The H0 is rejected because the p-value < α.
5. There appears to be a difference in the distribution of dolphin groups into each activity type among the four times of day. In fact, it appears that dolphins tend to socialize more in the mornings, travel during the noon and afternoon periods, and feed in the evenings (Table 20.6).
Table 20.6: The percentage of dolphin observations in each activity type by time of day.
Travel Feed Social Sum
Morning 8.3 38.9 52.8 100.0
Noon 38.5 26.9 34.6 100.0
Afternoon 57.7 3.8 38.5 100.0
Evening 16.5 70.9 12.7 100.1

1. Not needed for a chi-square test.

### R Appendix:

distrib(68.400,distrib="chisq",df=6,lower.tail=FALSE)

1. Subsequent sections demonstrate how this test statistic is used to compare observed frequencies (i.e., from a sample) to expected frequencies (i.e., from a null hypothesis).↩

2. How to find the df will be discussed in subsequent sections.↩

3. The Chi-Square Test is quite flexible and can be derived from different types of hypotheses than those described here.↩

4. Note a slight difference here because 0.363 was rounded to three decimals, whereas the fraction is not rounded.↩