text:
Abstract

The authors conducted a follow-up study of 261 women born during 1959–1965 (38% White, 40% African-American, and 22% Latina) to investigate whether maternal and infant factors are independently associated with adult body size after accounting for childhood growth. Standard statistical methods (linear regression and logistic regression) were compared with quantile regression methods to assess the independent effect maternal factors (body mass index (BMI; weight (kg)/height (m)2), maternal weight gain), birth measures (birth weight, placental weight), and early infancy and childhood growth measures (birth–4 months, 4 months–1 year, and 1–7 years) have on predicting adult body size. While most of these factors were important predictors of BMI at age 20 years, the size and relative importance of the effect differed across models. For example, maternal weight gain was associated with being overweight (BMI ≥ 25) at age 20 years (per 10-pound (4.5-kg) change, odds ratio = 1.65, 95% confidence interval: 1.11, 2.44) and was associated with the upper quantiles (≥75th percentile) of BMI at age 20 years. In contrast, maternal BMI and birth weight were relatively more important for lower quantiles, particularly at age 40 years. Only rapid growth from ages 1 to 7 years was an important predictor of adult BMI at both age 20 and age 40, irrespective of statistical model. However, the persistence of effects of maternal and infant factors on adult BMI, even after rapid childhood growth is accounted for, suggests a greater need to investigate these early-life influences and whether their impact differs for smaller and larger women.

Examining the role early-life factors may play in shaping adult body size is crucial to understanding human growth and preventing obesity. The positive correlation between birth weight and adult body size has motivated a search for the fetal origins of obesity (see reviews by Oken and Gillman (1) and Monteiro and Victora (2)). The findings of several large, well-conducted studies have supported a role for early-life factors in influencing adult body size (3–8). These studies illustrate the need to jointly examine maternal, infant, and childhood factors in order to estimate whether such factors operate independently or through other factors. For example, some studies with information on both birth weight and maternal body mass index (BMI; weight (kg)/height (m)2) have suggested that maternal BMI may be responsible for some of the previous correlations between birth weight and adult body size (3, 9).

Just as it is important to deconstruct birth weight and understand what it represents, it is also important to collect multiple measures of infant and childhood growth, because many maternal and infant factors probably influence adult body size through their impact on childhood growth. The results of two studies have supported a long-term effect of rapid growth during the first year of life (7, 8). If rapid infant growth predicts larger adult body size even after childhood growth patterns have been considered, it may be necessary to target multiple critical time points in long-term obesity prevention efforts.

Investigators in previous studies have examined the impact these early-life factors may have on adult body size primarily by considering their effect on mean adult BMI (linear regression) and/or the risk of being overweight (BMI ≥ 25) (logistic regression). We compared these standard methods with another method of estimation using quantile regression (10), which allows the effect of each factor to vary by the quantile of the response variable. The effects of maternal factors and early growth on persons who are underweight or overweight in adulthood may differ from those in the average cohort. Quantifying these differences, where they may exist, is crucial in designing health promotion interventions appropriate for different ranges of adult BMI and not just the average. We followed a birth cohort of women born in New York City from 1959 to 1965 in order to jointly examine the effect maternal factors (maternal BMI and pregnancy weight gain), birth measures (birth weight and placental weight), and infant and early childhood growth have on adult body size. We further examined whether the impact of these factors differs between smaller and larger women.

MATERIALS AND METHODS

Study participants

All women born at Columbia-Presbyterian Medical Center (New York, New York) from 1959 to 1965 who were included in the National Collaborative Perinatal Project until age 7 years were eligible for participation; there were 809 women eligible. Of the 340 participants (42 percent of the original cohort) we were able to trace, 77 percent participated in the adult follow-up (2001–2006); 38 percent were White, 40 percent were African-American, and 22 percent were Latina. The study was approved by the Internal Review Board at Columbia-Presbyterian Medical Center. Further details on the National Collaborative Perinatal Project are available elsewhere (11).

Growth data

Mothers were enrolled in the National Collaborative Perinatal Project during their second or third trimester of pregnancy. Prospective data on several growth measures, including birth weight, birth length, placental weight, and weight and height at ages 4 months, 1 year, 4 years, and 7 years, were obtained by trained clinical researchers using a standard protocol (11). Because the subjects did not all attend clinical examinations at exactly these time points, and to reduce any potential bias arising from the shifted measurement times, we interpolated these measurements at the target times using individual cubic interpolation splines. No interpolation was needed for birth measurements. Maternal weight gain and maternal prepregnancy BMI were based on weight and height measured prior to birth and reported weight prior to pregnancy, respectively.

We sent a questionnaire to all women who were successfully traced to obtain information on adult body size (height and weight at ages 20 years and 40 years) and other information about adult health and reproductive events.

Statistical analyses

Most studies of adult body size apply either linear regression methods with BMI entered as a continuous dependent variable or logistic regression methods with BMI entered as a dichotomized dependent variable (usually defined as overweight (BMI ≥ 25) vs. not overweight). These methods assume that the relation between independent variables and the dependent variable is the same irrespective of the scale of adult BMI. We used quantile regression (10) to estimate the effect of maternal, birth, and postnatal growth at various quantiles of the adult BMI distribution. Quantile regression aims at fully examining the effect of X on the entire distribution of Y, not just the mean value of Y. We selected the quantiles by internal percentile rank, rather than using absolute cutoff values.

We characterized an individual growth trajectory according to four parameters: birth measurement and percentile rank changes over three consecutive periods, from birth to age 4 months, from age 4 months to age 1 year, and from age 1 year to age 7 years. The changes in percentile rank provided us with a convenient and intuitive way to assess the growth rate while avoiding additional adjustment for age-dependent measurement scales. We defined three main patterns of growth: rapid, stable, and slow. Rapid growth defined participants whose percentile rank increased across two major reference percentiles of growth from birth to age 7 years. Reference percentiles were defined by standard Centers for Disease Control and Prevention growth charts as the 5th, 10th, 25th, 50th, 75th, and 95th percentiles. Stable growth defined participants whose percentile rank stayed within two major percentiles from birth to age 7 years. Slow growth defined participants whose percentile rank decreased across two major percentiles from birth to age 7 years.

We modeled the relation beween maternal factors (BMI and pregnancy weight gain), birth weight, and growth during the three time periods (birth–4 months, 4 months–1 year, and 1–7 years) using linear quantile regression models and standard methods (ordinary linear regression and logistic regression) for comparison. We also examined whether the following variables added to the overall goodness of fit of the base model (12, 13): length of gestation, preeclampsia, placental weight, race, socioeconomic status at birth, and childhood socioeconomic status measured at age 7 years. We examined the overall goodness of fit by calculating the relative deviance, which is the counterpart of R2 in ordinary linear mean regression (14). Finally, to study the long-term effect of the covariates, we applied the same models to BMI at age 40 years.

RESULTS

Table 1 shows descriptive statistics for the cohort. Women who participated in the adult follow-up did not differ from the overall cohort of eligible participants with regard to any of the infant and childhood growth measures. Table 2 shows the associations according to three percentiles (10th, 50th, and 90th) for three models examining the maternal variables (model 1), the maternal variables plus birth weight (model 2), and the maternal variables, birth weight, and postnatal growth variables (model 3). The parameter estimates differ across percentiles, suggesting that standard regression methods would not fully describe associations between these factors and adult body size. The parameter estimates are fairly stable across the three models for maternal BMI and maternal weight gain, suggesting that there is little mediation of these factors by birth weight and infant and childhood growth measures. Birth weight estimates, however, change significantly with the addition of the postnatal growth variables. Specifically, after the addition of the postnatal growth variables, the effect of birth weight is positive for the lower quantiles (≤50th percentile).

TABLE 1.

Characteristics of women in the New York Women's Birth Cohort (women born in 1959–1965) who participated in adult follow-up, 2001–2006

 Women in adult cohort (n = 261) All subjects eligible for adult follow-up (n = 809) Mean SD* % Median Mean SD Birth weight (g) 3,138.2 490.0 3,147.0 3,126.9 488.4 Placental weight (g) 452.4 92.7 450.0 447.0 92.0 Length of gestation (weeks) 39.5 2.6 40.0 39.3 3.0 Maternal prepregnancy BMI*,† 22.5 3.7 21.9 22.6 4.1 Maternal weight gain (pounds‡) during pregnancy 23.2 11.0 22.0 23.1 10.7 Weight (g) at age 4 months 6,103.2 782.6 6,095.0 6,138.0 776.8 Weight (kg) at age 1 year 9.6 1.1 9.6 9.69 1.26 Weight (kg) at age 4 years 17.0 2.8 16.3 17.2 3.1 Weight (kg) at age 7 years 23.9 4.9 22.8 24.3 5.3 Weight (kg) at age 20 years 58.7 12.2 55.8 NA* Height (m) at age 20 years 1.64 0.07 1.6 NA BMI at age 20 years 22.0 4.3 21.1 NA BMI <25 (normal weight) 87 BMI ≥25 (overweight) 13 BMI at age 40 years 27.2 6.9 25.8 NA BMI <25 (normal weight) 46 BMI ≥25 (overweight) 54
 Women in adult cohort (n = 261) All subjects eligible for adult follow-up (n = 809) Mean SD* % Median Mean SD Birth weight (g) 3,138.2 490.0 3,147.0 3,126.9 488.4 Placental weight (g) 452.4 92.7 450.0 447.0 92.0 Length of gestation (weeks) 39.5 2.6 40.0 39.3 3.0 Maternal prepregnancy BMI*,† 22.5 3.7 21.9 22.6 4.1 Maternal weight gain (pounds‡) during pregnancy 23.2 11.0 22.0 23.1 10.7 Weight (g) at age 4 months 6,103.2 782.6 6,095.0 6,138.0 776.8 Weight (kg) at age 1 year 9.6 1.1 9.6 9.69 1.26 Weight (kg) at age 4 years 17.0 2.8 16.3 17.2 3.1 Weight (kg) at age 7 years 23.9 4.9 22.8 24.3 5.3 Weight (kg) at age 20 years 58.7 12.2 55.8 NA* Height (m) at age 20 years 1.64 0.07 1.6 NA BMI at age 20 years 22.0 4.3 21.1 NA BMI <25 (normal weight) 87 BMI ≥25 (overweight) 13 BMI at age 40 years 27.2 6.9 25.8 NA BMI <25 (normal weight) 46 BMI ≥25 (overweight) 54
*

SD, standard deviation; BMI, body mass index; NA, not applicable.

Weight (kg)/height (m)2.

1 pound = 0.45 kg.

TABLE 2.

Association between early-life growth and maternal factors and body mass index at age 20 years, New York Women's Birth Cohort adult follow-up, 2001–2006

 Covariate Model 1† Model 2‡ Model 3§ 10th percentile 50th percentile 90th percentile 10th percentile 50th percentile 90th percentile 10th percentile 50th percentile 90th percentile β SE¶ β SE β SE β SE β SE β SE β SE β SE β SE Maternal body mass index# 0.15 0.09 0.21* 0.09 0.49* 0.20 0.16 0.11 0.21* 0.11 0.69* 0.28 0.18* 0.07 0.20 0.10 0.56** 0.21 Maternal weight gain (pounds††) during pregnancy −0.01 0.02 0.03 0.04 0.22** 0.09 −0.01 0.02 0.03 0.04 0.25** 0.08 −0.001 0.03 0.04 0.03 0.24** 0.07 Birth weight (kg) 0.19 0.48 0.08 0.85 −2.71 2.10 1.54** 0.47 1.85* 0.73 0.62 2.01 Postnatal growth rate (percentile change) Birth–age 4 months 0.03* 0.01 0.04** 0.01 0.05 0.03 Ages 4 months–1 year 0.03** 0.01 0.03** 0.01 0.06* 0.03 Ages 1–7 years 0.04** 0.01 0.06** 0.02 0.10** 0.03
 Covariate Model 1† Model 2‡ Model 3§ 10th percentile 50th percentile 90th percentile 10th percentile 50th percentile 90th percentile 10th percentile 50th percentile 90th percentile β SE¶ β SE β SE β SE β SE β SE β SE β SE β SE Maternal body mass index# 0.15 0.09 0.21* 0.09 0.49* 0.20 0.16 0.11 0.21* 0.11 0.69* 0.28 0.18* 0.07 0.20 0.10 0.56** 0.21 Maternal weight gain (pounds††) during pregnancy −0.01 0.02 0.03 0.04 0.22** 0.09 −0.01 0.02 0.03 0.04 0.25** 0.08 −0.001 0.03 0.04 0.03 0.24** 0.07 Birth weight (kg) 0.19 0.48 0.08 0.85 −2.71 2.10 1.54** 0.47 1.85* 0.73 0.62 2.01 Postnatal growth rate (percentile change) Birth–age 4 months 0.03* 0.01 0.04** 0.01 0.05 0.03 Ages 4 months–1 year 0.03** 0.01 0.03** 0.01 0.06* 0.03 Ages 1–7 years 0.04** 0.01 0.06** 0.02 0.10** 0.03
*

p ≤ 0.05;

**

p ≤ 0.01.

Model 1: maternal variables (body mass index and weight gain during pregnancy).

Model 2: the maternal variables plus birth weight.

§

Model 3: the maternal variables plus birth weight and postnatal growth variables (birth–4 months, 4 months–1 year, and 1–7 years).

SE, standard error.

#

Weight (kg)/height (m)2.

††

1 pound = 0.45 kg.

Table 3 presents results from the comparison of three statistical regression methods (quantile regression, ordinary linear regression, and logistic regression dichotomizing BMI into overweight (BMI ≥ 25) versus not overweight (BMI < 25)). Both linear and logistic regression models supported positive associations for maternal weight gain, birth weight, and rapid postnatal growth. The parameter estimates presented in the table reflect a one-unit increase in each covariate (weight gain in pounds, birth weight in kg, and all child growth measures expressed as percentile change for the time period). The last section of the table translates the logistic regression estimates into interpretable units, by variable. For example, maternal weight gain and postnatal growth were associated with being overweight (BMI ≥ 25) at age 20 years (per 10 pounds (4.5 kg) of maternal weight gain, odds ratio = 1.65, 95 percent confidence interval: 1.11, 2.44; per 10 percent change in rate of growth from ages 1 to 7 years, odds ratio = 1.73, 95 percent confidence interval: 1.36, 2.19). Estimates for all of the covariates differed by quantile of BMI, which cannot be observed using linear and logistic regression methods. For example, the impact of maternal BMI varied by quantile and was not significant in the logistic model. Maternal weight gain had a strong association with BMI at age 20 years for the upper quantiles (≥75th percentile), but the association was underestimated using ordinary linear regression. Birth weight was not associated with BMI at age 20 years among larger women (90th percentile). Growth during the first 4 months of life was associated with BMI at age 20 years for all percentiles but the 90th. Rapid growth from ages 1 to 7 years was associated with adult BMI for all percentiles.

TABLE 3.

Associations of maternal factors, birth weight, and early-life growth with body mass index at age 20 years, New York Women's Birth Cohort adult follow-up, 2001–2006

 Quantile† Model 10th percentile 25th percentile 50th percentile 75th percentile 90th percentile Linear model‡ Logistic model§ β SE¶ Rank# β SE Rank# β SE Rank# β SE Rank# β SE Rank# β SE β SE OR¶ 95% CI¶ Increment of change for OR Maternal body mass index†† 0.18* 0.07 1 0.14* 0.07 1 0.19 0.11 2 0.36** 0.10 2 0.56** 0.20 3 0.30** 0.08 0.12 0.07 1.13 0.98, 1.29 1 unit Maternal weight gain (pounds‡‡) during pregnancy −0.002 0.03 6 0.02 0.03 6 0.04 0.03 5 0.13** 0.04 3 0.24** 0.06 2 0.10** 0.03 0.05* 0.02 1.65 1.11, 2.44 10 pounds (4.5 kg) Birth weight (kg) 1.54** 0.48 4 1.71** 0.62 4 1.85* 0.78 6 1.82* 0.90 6 0.62 1.89 6 1.47* 0.73 2.65** 0.78 3.76 1.75, 8.08 500 g Postnatal growth rate (percentile change) Birth–age 4 months 0.03** 0.01 3 0.04** 0.01 3 0.04** 0.01 3 0.05* 0.02 4 0.05 0.03 4 0.05** 0.01 0.05** 0.01 1.65 1.30, 2.09 10% change Ages 4 months–1 year 0.03** 0.01 5 0.02 0.01 5 0.03** 0.01 4 0.05** 0.01 5 0.06* 0.03 5 0.05** 0.01 0.04** 0.01 1.48 1.13, 1.93 10% change Ages 1–7 years 0.03** 0.01 2 0.05** 0.01 2 0.06** 0.01 1 0.07** 0.02 1 0.10** 0.03 1 0.07** 0.01 0.05** 0.01 1.73 1.36, 2.19 10% change Goodness of fit ($τ^$ − τ) 0.00 0.01 0.00 0.02 0.01
 Quantile† Model 10th percentile 25th percentile 50th percentile 75th percentile 90th percentile Linear model‡ Logistic model§ β SE¶ Rank# β SE Rank# β SE Rank# β SE Rank# β SE Rank# β SE β SE OR¶ 95% CI¶ Increment of change for OR Maternal body mass index†† 0.18* 0.07 1 0.14* 0.07 1 0.19 0.11 2 0.36** 0.10 2 0.56** 0.20 3 0.30** 0.08 0.12 0.07 1.13 0.98, 1.29 1 unit Maternal weight gain (pounds‡‡) during pregnancy −0.002 0.03 6 0.02 0.03 6 0.04 0.03 5 0.13** 0.04 3 0.24** 0.06 2 0.10** 0.03 0.05* 0.02 1.65 1.11, 2.44 10 pounds (4.5 kg) Birth weight (kg) 1.54** 0.48 4 1.71** 0.62 4 1.85* 0.78 6 1.82* 0.90 6 0.62 1.89 6 1.47* 0.73 2.65** 0.78 3.76 1.75, 8.08 500 g Postnatal growth rate (percentile change) Birth–age 4 months 0.03** 0.01 3 0.04** 0.01 3 0.04** 0.01 3 0.05* 0.02 4 0.05 0.03 4 0.05** 0.01 0.05** 0.01 1.65 1.30, 2.09 10% change Ages 4 months–1 year 0.03** 0.01 5 0.02 0.01 5 0.03** 0.01 4 0.05** 0.01 5 0.06* 0.03 5 0.05** 0.01 0.04** 0.01 1.48 1.13, 1.93 10% change Ages 1–7 years 0.03** 0.01 2 0.05** 0.01 2 0.06** 0.01 1 0.07** 0.02 1 0.10** 0.03 1 0.07** 0.01 0.05** 0.01 1.73 1.36, 2.19 10% change Goodness of fit ($τ^$ − τ) 0.00 0.01 0.00 0.02 0.01
*

p ≤ 0.05;

**

p ≤ 0.01.

Parameter estimates and standard errors based on quantile regression methods, by quantile.

Parameter estimates and standard errors from ordinary least-squares regression.

§

Parameter estimates and standard errors from a logistic model comparing body mass index ≥ 25 with body mass index < 25.

#

Ranking of relative predictive importance based on relative deviance.

SE, standard error; OR, odds ratio; CI, confidence interval.

††

Weight (kg)/height (m)2.

‡‡

1 pound = 0.45 kg.

Table 3 also shows the overall goodness of fit for the quantile regression model. The difference between the model-based empirical distribution and its nominal values ranged from 0.00 to 0.02, suggesting that the model fitted the data very well. After adjustment for the factors listed in table 3, race, socioeconomic score at birth and at age 7 years, length of gestation, preeclampsia, placental weight, and maternal smoking status did not statistically improve the prediction of BMI at age 20 years. Table 3 also ranks the covariates in order of importance in overall model fitness, on the basis of deviance statistics (with a lower ranking indicating more importance in predicting BMI). Although all of the variables were important predictors, the relative ranking suggested that rapid growth from ages 1 to 7 years, maternal BMI, and rapid growth during the first 4 months of life had the most predictive power for the lower quantiles (≤50th percentile), whereas rapid growth from ages 1 to 7 years, maternal BMI, and maternal weight gain had the most predictive power for the upper quantiles of BMI at age 20 years (≥75th percentile).

Figures 1 and 2 illustrate the estimated conditional distributions of adult BMI given combinations of pre- and postnatal factors. The x-axis of each graph shows the quantile level, and the y-axis shows the corresponding adult BMI. In each panel, we altered one risk factor while maintaining the others at constant levels, which enabled us to study the impact of that factor on the entire distribution of adult BMIs. BMIs of 25 and 30 represent the cutoff values used to define overweight and obesity, respectively.

FIGURE 1.

Association between maternal weight gain during pregnancy and offspring's body mass index (BMI; weight (kg)/height (m)2) at age 20 years, conditional on stable postnatal growth, New York Women's Birth Cohort (women born in 1959–1965). The x-axis represents the quantile level (e.g., 0.2 = 20th percentile) of offspring BMI, and the y-axis represents the corresponding predicted BMI at age 20 years based on the quantile-specific parameter estimates and the following covariate settings: stable postnatal growth (no change in percentile rank), median birth weight (3.2 kg), and median maternal prepregnancy BMI (21.9). Maternal weight gain varies as follows: panel A—10th percentile of maternal weight gain (11 pounds (5 kg)); panel B—50th percentile of maternal weight gain (22 pounds (10 kg)); panel C—90th percentile of maternal weight gain (37.4 pounds (17 kg)). The dotted lines represent pointwise 95 percent confidence intervals.

FIGURE 1.

Association between maternal weight gain during pregnancy and offspring's body mass index (BMI; weight (kg)/height (m)2) at age 20 years, conditional on stable postnatal growth, New York Women's Birth Cohort (women born in 1959–1965). The x-axis represents the quantile level (e.g., 0.2 = 20th percentile) of offspring BMI, and the y-axis represents the corresponding predicted BMI at age 20 years based on the quantile-specific parameter estimates and the following covariate settings: stable postnatal growth (no change in percentile rank), median birth weight (3.2 kg), and median maternal prepregnancy BMI (21.9). Maternal weight gain varies as follows: panel A—10th percentile of maternal weight gain (11 pounds (5 kg)); panel B—50th percentile of maternal weight gain (22 pounds (10 kg)); panel C—90th percentile of maternal weight gain (37.4 pounds (17 kg)). The dotted lines represent pointwise 95 percent confidence intervals.

FIGURE 2.

Association between postnatal growth velocity and body mass index (BMI; weight (kg)/height (m)2) at age 20 years, conditional on birth weight, New York Women's Birth Cohort (women born in 1959–1965). The x-axis represents the quantile level (e.g., 0.2 = 20th percentile) of BMI, and the y-axis represents the corresponding predicted BMI at age 20 years based on the quantile-specific parameter estimates and the following covariate settings. Panels A–C show results from models conditional on low birth weight (10th percentile = 2.5 kg) and median maternal prepregnancy BMI (21.8) and median maternal weight gain (22 pounds (10 kg)) during pregnancy and three types of postnatal growth: stable postnatal growth (panel A), rapid postnatal growth from birth to age 4 months (panel B), and rapid postnatal growth from ages 1 to 7 years (panel C). Panels D–F show results from models conditional on high birth weight (90th percentile = 3.7 kg) and median maternal BMI (21.8) and median maternal weight gain (22 pounds (10 kg)) and three types of postnatal growth: stable postnatal growth (panel D), slow postnatal growth from birth to age 4 months (panel E), and slow postnatal growth from ages 1 to 7 years (panel F). The dotted lines represent pointwise 95 percent confidence intervals.

FIGURE 2.

Association between postnatal growth velocity and body mass index (BMI; weight (kg)/height (m)2) at age 20 years, conditional on birth weight, New York Women's Birth Cohort (women born in 1959–1965). The x-axis represents the quantile level (e.g., 0.2 = 20th percentile) of BMI, and the y-axis represents the corresponding predicted BMI at age 20 years based on the quantile-specific parameter estimates and the following covariate settings. Panels A–C show results from models conditional on low birth weight (10th percentile = 2.5 kg) and median maternal prepregnancy BMI (21.8) and median maternal weight gain (22 pounds (10 kg)) during pregnancy and three types of postnatal growth: stable postnatal growth (panel A), rapid postnatal growth from birth to age 4 months (panel B), and rapid postnatal growth from ages 1 to 7 years (panel C). Panels D–F show results from models conditional on high birth weight (90th percentile = 3.7 kg) and median maternal BMI (21.8) and median maternal weight gain (22 pounds (10 kg)) and three types of postnatal growth: stable postnatal growth (panel D), slow postnatal growth from birth to age 4 months (panel E), and slow postnatal growth from ages 1 to 7 years (panel F). The dotted lines represent pointwise 95 percent confidence intervals.

Figure 1 shows the association between maternal weight gain and BMI at age 20 years, conditional on stable postnatal growth, median birth weight (3.2 kg), and median maternal BMI (21.9), obtained using the parameter estimates from table 3. Women born to mothers who gained more weight during pregnancy (90th percentile (panel C)) were more likely to be overweight (BMI ≥ 25) at age 20 years than women whose mothers gained less weight during pregnancy (10th (panel A) and 50th (panel B) percentiles).

Figure 2 shows the combined effects of birth weight and postnatal growth on adult BMI. Conditional on low birth weight, participants who experienced rapid growth from ages 1 to 7 years (panel C) were more likely to be overweight at age 20 years than those who experienced rapid growth from birth to age 4 months (panel B) or stable postnatal growth (panel A). Conditional on high birth weight, participants who experienced stable postnatal growth (panel D) were more likely to be overweight at age 20 years than those who experienced slower growth from birth to age 4 months (panel E). Panel F shows that large babies who experienced slower growth from ages 1 to 7 years were unlikely to be overweight at age 20 years. Thus, low birth weight babies who experienced rapid growth from ages 1 to 7 years were the most likely to be overweight at age 20 years.

Table 4 shows the relations with BMI at age 40 years. Rapid growth from ages 1 to 7 years was the only variable that was important for all quantiles of BMI at age 40 years. Maternal BMI was important for all percentiles but the 90th. Maternal weight gain and birth weight were no longer significant in the prediction of BMI at age 40 years at any quantile level. In addition, the variability of the parameter estimates increased, particularly for birth weight, but the overall fit of the model remained good. The linear and logistic models also supported an association between maternal BMI and growth from ages 1 to 7 years and BMI at age 40 years. In addition, the linear model supported an association between birth weight and rapid growth from ages 1 to 4 months. The logistic model did not indicate any relation with birth weight but supported an association between rapid growth during the first 4 months of life and BMI at age 40 years.

TABLE 4.

Associations of maternal factors, birth weight, and early-life growth with body mass index at age 40 years, New York Women's Birth Cohort adult follow-up, 2001–2006

 Quantile† Model 10th percentile 25th percentile 50th percentile 75th percentile 90th percentile Linear model‡ Logistic model§ β SE¶ Rank# β SE Rank# β SE Rank# β SE Rank# β SE Rank# β SE β SE OR¶ 95% CI¶ Increment of change for OR Maternal body mass index†† 0.31** 0.11 1 0.36** 0.11 1 0.26 0.15 1 0.37* 0.17 2 0.29 0.31 2 0.38** 0.13 0.12* 0.05 1.13 1.03, 1.24 1 unit Maternal weight gain (pounds‡‡) during pregnancy 0.04 0.05 3 0.02 0.04 4 0.08 0.05 4 0.14 0.08 3 0.21 0.12 3 0.07 0.04 0.03 0.02 1.30 0.96, 1.77 10 pounds (4.5 kg) Birth weight (kg) 1.29 1.15 6 0.48 1.15 6 1.72 1.43 5 2.24 1.70 5 4.96 2.62 4 2.50* 1.19 0.38 0.41 1.21 0.81, 1.80 500 g Postnatal growth rate (percentile change) Birth–age 4 months 0.05* 0.02 4 0.02 0.02 3 0.04 0.02 3 0.07* 0.03 3 0.00 0.04 6 0.04* 0.02 0.01* 0.01 1.16 1.01, 1.33 10% change Ages 4 months–1 year 0.04* 0.02 5 0.01 0.02 5 0.01 0.03 6 0.03 0.02 6 0.04 0.04 5 0.03 0.02 0.01 0.01 1.10 0.95, 1.27 10% change Ages 1–7 years 0.05* 0.02 2 0.05** 0.02 2 0.09** 0.03 2 0.12** 0.02 1 0.18** 0.05 1 0.10** 0.02 0.03** 0.01 1.28 1.12, 1.46 10% change Goodness of fit ($τ^$ − τ) 0.03 0.02 −0.03 −0.02 0.00
 Quantile† Model 10th percentile 25th percentile 50th percentile 75th percentile 90th percentile Linear model‡ Logistic model§ β SE¶ Rank# β SE Rank# β SE Rank# β SE Rank# β SE Rank# β SE β SE OR¶ 95% CI¶ Increment of change for OR Maternal body mass index†† 0.31** 0.11 1 0.36** 0.11 1 0.26 0.15 1 0.37* 0.17 2 0.29 0.31 2 0.38** 0.13 0.12* 0.05 1.13 1.03, 1.24 1 unit Maternal weight gain (pounds‡‡) during pregnancy 0.04 0.05 3 0.02 0.04 4 0.08 0.05 4 0.14 0.08 3 0.21 0.12 3 0.07 0.04 0.03 0.02 1.30 0.96, 1.77 10 pounds (4.5 kg) Birth weight (kg) 1.29 1.15 6 0.48 1.15 6 1.72 1.43 5 2.24 1.70 5 4.96 2.62 4 2.50* 1.19 0.38 0.41 1.21 0.81, 1.80 500 g Postnatal growth rate (percentile change) Birth–age 4 months 0.05* 0.02 4 0.02 0.02 3 0.04 0.02 3 0.07* 0.03 3 0.00 0.04 6 0.04* 0.02 0.01* 0.01 1.16 1.01, 1.33 10% change Ages 4 months–1 year 0.04* 0.02 5 0.01 0.02 5 0.01 0.03 6 0.03 0.02 6 0.04 0.04 5 0.03 0.02 0.01 0.01 1.10 0.95, 1.27 10% change Ages 1–7 years 0.05* 0.02 2 0.05** 0.02 2 0.09** 0.03 2 0.12** 0.02 1 0.18** 0.05 1 0.10** 0.02 0.03** 0.01 1.28 1.12, 1.46 10% change Goodness of fit ($τ^$ − τ) 0.03 0.02 −0.03 −0.02 0.00
*

p ≤ 0.05;

**

p ≤ 0.01.

Parameter estimates and standard errors based on quantile regression methods, by quantile.

Parameter estimates and standard errors from ordinary least-squares regression.

§

Parameter estimates and standard errors from a logistic model comparing body mass index ≥ 25 with body mass index < 25.

#

Ranking of relative predictive importance based on relative deviance.

SE, standard error; OR, odds ratio; CI, confidence interval.

††

Weight (kg)/height (m)2.

‡‡

1 pound = 0.45 kg.

DISCUSSION

We found that maternal BMI, maternal weight gain, birth weight, and rapid childhood growth had long-lasting effects on adult body size in women but that the impact of these factors differed between larger and smaller adult women. Rapid growth from ages 1 to 7 years and maternal prepregnancy BMI were the most important predictors for all percentiles of age-20 BMI except the 90th percentile, where maternal weight gain was more important than maternal BMI. Rapid growth from ages 1 to 7 years and maternal prepregnancy BMI were the most important factors predicting BMI at age 40 years. Stettler et al. have argued that rapid growth during the first few months, or even weeks, of life may be associated with greater childhood (15) and adult (8) obesity. We also observed a strong association between rapid growth from birth to age 4 months and BMI at age 20 years for all percentiles below the 90th. Interestingly, rapid growth during the first 4 months of life was also associated with BMI at age 40 years in both linear and logistic models and was important for the 10th and 75th percentiles of BMI at age 40 years. This suggests that while rapid childhood growth from ages 1 to 7 years is most important in predicting adult BMI, early infant growth is also independently predictive of adult BMI.

The strong association between maternal prepregnancy BMI and adult body size may reflect some combination, as Whitaker and Dietz (16) and others have argued, of transmitted genes as well as environmental influences (17, 18). Even after accounting for maternal prepregnancy BMI, we found a strong association between maternal weight gain and body size at age 20 years among larger women (≥75th percentile). This observation suggests that pregnancy-specific weight gain may have a persistent long-term impact on the offspring's adult BMI. Thus, our study supports the work of Dietz (19), which suggested that there are critical periods influencing adult body size, including the intrauterine period, early infancy, and early childhood. While we did not have data on body size during adolescence, our observed and predicted values of adult BMI were very close, suggesting that the unexplained portion of the model was small.

After we accounted for both postnatal growth rates and maternal variables, birth weight was still associated with body size at age 20 years among all but the largest women (≤75th percentile). However, we did not find that this association held for larger women—an observation that would be lost without the quantile-specific estimates. Our data help to illustrate why studies examining just birth weight and adult body size are incomplete, because the effect of birth weight may be different among smaller and larger women.

In this study, we wanted to examine whether pre- and postnatal factors were associated with adult BMI and, further, whether these factors were similar across quantiles of BMI. We did not assume that the influences of these factors were the same for larger and smaller women. Just as ordinary mean regression estimates the influence of risk factors on the “population average,” quantile regression allows for complete examination of their influences on the entire distribution of adult BMIs. If our estimates were the same for all quantiles of adult BMI, then ordinary linear regression would be appropriate. Logistic regression is flexible and easy to interpret, but it requires a subjective choice of the threshold. For example, the cutpoint used to define overweight on the basis of BMI has changed over time (20). In contrast, in quantile regression, the overweight and underweight cutpoints are determined by internal percentile ranks.

We considered other methods, such as nonlinear regression, which models the mean of the response variable (Y) as a nonlinear function of the covariates (X). In contrast, quantile regression aims at fully examining the effect of X on the entire distribution of Y, not just on the mean value of Y. Differences across quantile levels of Y do not imply that the mean of Y is nonlinear. The effect on the quantile level could be either linear or nonlinear. This method of estimation has already been applied in a number of health-related investigations (21–27). Thus, by modeling our data with quantile regression, as well as comparing our estimates with the results of more standard approaches, we were able to examine whether pre- and postnatal factors differed for smaller and larger women. Differences by quantiles suggest that factors may differentially affect the adipose/lean tissue proportion and may suggest underlying pathways linking these observations.

Our findings are unlikely to have been influenced by selection bias, because subjects who were lost to follow-up did not differ from subjects who were not lost according to any of the growth or maternal factors examined. Data on all infant and childhood measures were collected prospectively using a uniform measurement protocol. Nondifferential measurement error in adult BMI would lead to attenuated associations and therefore cannot explain the findings that we report here. However, if adult women differentially reported their weight such that overweight women were more likely to underreport than average-weight women, we would expect to see even stronger correlations given the positive correlation between childhood weight and adult weight. We were able to assess confounding by a number of factors, including socioeconomic status at birth and age 7 years. We also assessed other important growth measures, including placental weight, birth length, and height changes during infancy and early childhood. These other measures did not improve the overall fit of the model, so we did not include them in our final model. In addition, other maternal obstetrical conditions such as preeclampsia and hypertension did not improve the fit of the overall model. Too few mothers had gestational diabetes for assessment of the impact of this condition on the growth parameters.

Our study suggests that adult body size is influenced by high maternal BMI and rapid postnatal growth. Even after these factors have been accounted for, birth weight is still an important predictor of adult BMI among smaller women and maternal weight gain is an important predictor among larger women. There is no reason to expect that factors should operate the same way across the continuum of adult BMI. Thus, is it possible, and plausible, that these relations may operate differently for larger and smaller women. Understanding the biologic relations underpinning these observations is paramount, since childhood obesity prevalence (28), birth weight (29), and maternal weight gain during pregnancy (30) are all increasing. For example, the median weight gain in the United States in 2003 was 30.5 pounds (13.9 kg) (30), 8.5 pounds (3.9 kg) more than the median in our cohort; our models suggest that this change alone would lead to an increase of more than two points in BMI at age 20 years among larger women, or a 50 percent overall increase in the risk of being overweight. Because the average age at pregnancy is also increasing, and BMI increases with age in most populations, maternal prepregnancy BMI is probably higher as well. If our findings are replicated, these trends all point to dramatic long-term consequences for the prevalence of overweight in adulthood.

Abbreviations

Abbreviations
• BMI

body mass index

This study was funded by grants DAMD170210357 and K07CA90685 from the US Department of Defense Breast Cancer Research Program and the National Cancer Institute, respectively.

The authors thank the following persons for their contributions to the National Collaborative Perinatal Project Adult Follow-up Study: Dr. Ezra Susser, Dr. Parisa Teranifar, Julie Flom, Tamarra James, Lina Titievsky-Konikov, Dipal Shah, Jennifer Ferris, Tara Kalra, Shobana Ramachandran, Julia Meurling, Sujata Narayan, Summer Wright, and Adey Tsega. Conflict of interest: none declared.

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Author notes

Editor's note:An invited commentary on this article appears on page 14, and the authors' response appears on page 17.


text: it is symmetric and bell-shaped. Examples are the mean $$\mu = E(X)$$ of the distribution, the standard deviation $$\sigma = SD(X)$$, or a population proportion $$p$$. $${\sigma _{\bar X}} = \sqrt {\sum {{\bar X}^2}\,f\left( {\bar X} \right) – {{\left[ {\sum \bar X\,f\left( {\bar X} \right)} \right]}^2}} \,\,\,\, = \,\,\,\sqrt {\frac{{997}}{{36}} – {{\left( {\frac{{63}}{{12}}} \right)}^2}} = 0.3632$$. Find the mean and standard deviation of the sample mean. The mean and variance of the population are: $$\mu = \frac{{\sum X}}{N} = \frac{{45}}{5} = 9$$ and $${\sigma ^2} = \frac{{\sum {X^2}}}{N} – {\left( {\frac{{\sum X}}{N}} \right)^2} = \frac{{495}}{5} – {\left( {\frac{{45}}{5}} \right)^2} = 99 – 81 = 18$$, (i) $$E\left( {\bar X} \right) = \mu = 9$$ (ii) $${\text{Var}}\left( {\bar X} \right) = \frac{{{\sigma ^2}}}{n}\left( {\frac{{N – n}}{{N – 1}}} \right) = \frac{{18}}{2}\left( {\frac{{5 – 2}}{{5 – 1}}} \right) = 6.75$$. The sampling distribution is the distribution of the values that a statistic takes on different samples of the same size and from the same population. The central limit theorem states that whenever a random sample of size n is taken from any distribution with mean and variance, then the sample mean will be approximately normally distributed with mean and variance. We have that: Secondly, as we increase the sample size from 6 to 24, there appears to be a decrease in the variability of sample means (compare the variability in the vertical bars in panel (a) and panel(b)). If you sample one number from a standard normal distribution, what is the probability it … Binomial Distribution Plot 10+ Examples of Binomial Distribution. We use uppercase letters when we want to study the effects of sampling variation on a statistic, while we use lowercase letters for observed values. Hence state and verify relation between (a). Figure 6.1 Distribution of a Population and a Sample Mean. Thus, the number of possible samples which can be drawn without replacement is. Extract from the hollywood tibble the three variables of interest (Movie, Genre, Budget) and keep the movies for which we have all information (no missing entries). A certain population is strongly skewed to the left. $the population standard deviation divided by $$\sqrt{n}$$ with $$n$$ being the sample size. Because $$\sqrt{4} = 2$$ we halve $$\sigma_{\bar X}$$ by making the sample size 4 times as large. Example of Sampling Distribution Assuming that a researcher is conducting a study on the weights of the inhabitants of a particular town and he has five observations or samples, i.e., 70kg, 75kg, 85kg, 80kg, and 65kg. How systematic sampling works. Describe the sampling distribution of sample proportion by stating its mean, variance, and … The distribution of sample means computed by Mary and Alex are shown in the dotplot below in green and red, respectively.$. Understanding Sampling Distribution . If you can, it is best to measure the entire population. Hence, the standard deviation of the sample mean is called the standard error of the mean. In Figure 1 we display the individual gestation periods in each sample as dots, along with the means gestation period $$\bar x$$ of each sample. So, the standard error of the mean can be either computed as the standard deviation of the sampling distribution, or using the formula September 10 @ The distribution of sample means, or the sampling distribution, can help us understand this variability. \sigma_{\bar X} &= \frac{\sigma}{\sqrt{n}} = \frac{\text{Population standard deviation}}{\sqrt{\text{Sample size}}} Hence, a statistic is a numerical summary of a random experiment and for this reason it is a random variable, e.g. The standard error of a statistic, denoted $$SE$$, is the standard deviation of its sampling distribution. What is the distinction between an estimate and an estimator? Your email address will not be published. The random variable $$\bar X$$ follows a normal distribution: problems included are about: probabilities, mutually exclusive events and addition formula of probability, combinations, binomial distributions, normal distributions, reading charts. What is an estimate of the proportion of comedy movies using a sample of size 20? Form a sampling distribution of sample means. • Sampling distribution of the mean: probability distribution of ... • Example: All possible samples of size 10 from a class of 90 = 5.72*1012. Please tell me this question as soon as possible, Aimen Naveed \sigma_{\bar X} &= \frac{\sigma}{\sqrt{n}} = \frac{\text{Population standard deviation}}{\sqrt{\text{Sample size}}} Numerical summaries of that distribution are called parameters. \], https://uoepsy.github.io/data/pregnancies.csv, Harry Potter and the Order of the Phoenix, Recognise the difference between parameters and statistics, Be able to use a sample statistic to estimate an unknown parameter, Understand what a sampling distribution is. We can think of random or unpredictable data as arising in two ways. The variability, or spread, of the sampling distribution shows how much the sample statistics tend to vary from sample to sample. Speciﬁcally, it is the sampling distribution of the mean for a sample size of 2 (N = 2). Hence, they follow the shape of the normal curve. Suppose we take samples of size 1, 5, 10, or 20 from a population that consists entirely of the numbers 0 and 1, half the population 0, half 1, so that the population mean is 0.5. What is a parameter? The answers to these problems are at the bottom of the page. Imagine an urn with tickets, where each ticket has the name of each population unit. EXAMPLE: SAT MATH SCORES Take a sample of 10 random students from a population of 100. Check that the data were read into R correctly. \begin{aligned} For this simple example, the distribution of pool balls and the sampling distribution are both discrete distributions. X-, the mean of the measurements in a sample of size n; the distribution of X-is its sampling distribution, with mean μ X-= μ and standard deviation σ X-= σ / n. Example 3 Let X - be the mean of a random sample of size 50 drawn from a population with … Among the recorded variables, three will be of interest: Read the Hollywood movies data into R, and call it hollywood. What would the sampling distribution of the mean look like if we could afford to take samples as big as the entire population, i.e. The larger the value of the sample size, the better the approximation to the normal. Figure 5: Sampling distribution of the proportion for $$n = 20$$ with population parameter $$p$$ marked by a red vertical line. Remember that mutate() takes a tibble and adds or changes a column. the mean denoted $$\bar X$$. A parameter is a numerical summary of a population or distribution, for example the average income in the whole population. I. Its government has data on this entire population, including the number of times people marry. By taking multiple samples of size equal to the entire population, every time we would obtain the population parameter exactly, so the distribution would look like a histogram with a single bar on top of the true value: we would find the true parameter with a probability of one, and the estimation error would be 0. Increasing the sample size, the spread of the statistic values is reduced. Figure 6: Three sampling distributions of the proportion, with population parameter $$p$$ marked by a red vertical line. 12:23 pm, Draw all possible sample of size n = 3 with replacement from the population 3,6,9 and 12. We then increased the sample size to 24 women and took 12 samples each of 24 individuals. The value of a sample statistic such as the sample mean (X) is likely to be different for each sample that is drawn from a population. the estimate) is equal to the population mean (i.e. Each of the density histograms above displays the distribution of the sample mean, computed on samples of the same size and from the same population. It can be considered as the entire population of movies produced in Hollywood in that time period. We denote the estimate (observed value) with a lowercase letter and the estimator (random variable) with an uppercase letter. Compute the sampling distribution for the proportion of comedy movies using 1,000 samples each of size $$n = 20$$, $$n = 50$$, and $$n = 200$$ respectively. Can we compute the parameters within the next 30 minutes? • You might get a mean of 502 for that sample. Variance of the sampling distribution of the mean and the population variance. We call “estimate” the value of a statistic which is used to estimate an unknown population parameter. To get the function in your computer, run this code: into each file where you want to use the rep_sample_n() function. Because the catalogue has so many pages, we can not compute the population parameters within the next 30 minutes. An example could be average blood pressure in Scotland, or the proportion of people with a car. The sampling distributions are: n = 1: (6.2.2) x ¯ 0 1 P ( x ¯) 0.5 0.5. n = 5: A statistic is a numerical summary of the sample data. Draw all possible samples of size 2 without replacement from a population consisting of 3, 6, 9, 12, 15. If this is the quantity we are interested in, the obvious approach would be to take a sample from that population and use the proportion vaccinated in the sample, $$\hat{p}$$, as an estimate of $$p$$. From the above tibble we see that action movies have been allocated a higher budget ($$\mu_{Action} =$$ 85.9) than comedy movies ($$\mu_{Comedy} =$$ 36.9). Your Stat Class is the #1 Resource for Learning Elementary Statistics. Students in the library perhaps tend to study more. In practice, we know very little about the population we are sampling from (or the random process generating our data) and we collect data to find out more about these populations. The size of the sample is at 100 with a mean weight of 65 kgs and a standard deviation of 20 kg. The Greek letter $$\sigma$$ (sigma) represents the population standard deviation (parameter), while $$s$$ or $$\hat{\sigma}$$ (sigma-hat) is the standard deviation computed from the sample data (sample statistic). \bar X \sim N(\mu,\ SE) There's an island with 976 inhabitants. The process of sampling $$n$$ people from the population is a random process. (No bias), Shape: For most of the statistics we consider, if the sample size is large enough, the sampling distribution will follow a normal distribution, i.e. Mary sampled the students at random, while Alex asked students from the library. Here are some examples of Binomial distribution: Rolling a die: Probability of getting the number of six (6) (0, 1, 2, 3…50) while rolling a die 50 times; Here, the random variable X is the number of “successes” that is the number of times six occurs. Example: Means in quality control An auto-maker does quality control tests on the paint thickness at different points on its car parts since there is some variability in the painting process. the parameter), the sample mean $$\bar X$$ is an unbiased estimator of the population mean. The Central Limit Theorem is the sampling distribution of the sampling means approaches a normal distribution as the sample size gets larger, no matter what the shape of the data distribution. September 18 @ Therefore the parameters of interest are unknown quantities that we want to estimate. The mean and standard deviation of the population are: $$\mu = \frac{{\sum X}}{N} = \frac{{21}}{4} = 5.25$$ and $${\sigma ^2} = \sqrt {\frac{{\sum {X^2}}}{N} – {{\left( {\frac{{\sum X}}{N}} \right)}^2}} = \sqrt {\frac{{115}}{4} – {{\left( {\frac{{21}}{4}} \right)}^2}} = 1.0897$$, $$\frac{\sigma }{{\sqrt n }}\sqrt {\frac{{N – n}}{{N – 1}}} = \frac{{1.0897}}{{\sqrt 3 }}\sqrt {\frac{{4 – 3}}{{4 – 1}}} = 0.3632$$, Hence $${\mu _{\bar X}} = \mu$$ and $${\sigma _{\bar X}} = \frac{\sigma }{{\sqrt n }}\sqrt {\frac{{N – n}}{{N – 1}}}$$, Pearl Lamptey This tendency to overestimate the population parameter shows that the sampling method is biased. are actually samples, not populations. The arithmetic mean is 14.0 inches, and … Suppose we take samples of size 1, 5, 10, or 20 from a population that consists entirely of the numbers 0 and 1, half the population 0, half 1, so that the population mean is 0.5. You might remember it from the cartoon Wile E. Coyote and the Road Runner.↩, $$\sigma_{\bar X} = \frac{\sigma}{\sqrt{n}}$$, $Take all possible samples of size 3 with replacement from population comprising 10 12 14 16 18 make sampling distribution and verify, Aimen Naveed Thus, the number of possible samples which can be drawn without replacement is \left( {\begin{array}{*{20}{c}} N \\ n \end{array}} \right) = \left( {\begin{array}{*{20}{c}} 4 \\ 3 \end{array}} \right) = 4, {\mu _{\bar X}} = \sum \bar X\,f\left( {\bar X} \right)\,\,\,\, = \,\,\,\frac{{63}}{{12}} = 5.25$, $MichaelExamSolutionsKid 2016-09-08T21:29:50+00:00 Fig 1. Why did Mary and Alex get so different results? What do you notice in the distributions above? We notice that Alex got consistently higher estimates of the population mean study time than Mary did. Word Problem #3 (Normal Distribution) - SOLUTION Answer: .3483 Easy Solution: The solution to this problem requires noticing that the random variable is X, so that the standardization to Z must use the SE of X = σ / √n. SE = \sigma_{\bar X} = \frac{\sigma}{\sqrt{n}} We make the distinction because we refer to the random variable (estimator) when we want to study the variability of the statistic from sample to sample, for example to investigate how precise it is. We also notice that the density histograms in Figure 2 are symmetric and bell-shaped. From the above discussion, you can see that the population parameter and the sample statistic generally have the same name. You can inspect the sample data in the following interactive table in which the data corresponding to each sample have been colour-coded so that you can distinguish the rows belonging to the 1st, 2nd, …, and 12th sample: Now, imagine computing the mean of the six observation in each sample. The standard deviation of the sample means tells us that the variability in the sample means gets smaller smaller as the sample size increases. Using the appropriate notation, report your results in one or two sentences. There is an interesting patter in the decrease, which we will now verify. \frac{\bar X - \mu}{SE} \sim N(0, 1) We have taken multiple samples to show how the estimation error varies with the sample size. Form the sampling distribution of sample means and verify the results. \mu_{\bar X} &= \mu = \text{Population mean} \\ 9. On the other hand, Alex selected the most readily available people and took convenience samples. a sample) and use the proportion of faulty light bulbs in the sample $$\hat p$$ as an estimate of the underlying proportion of faulty light bulbs $$p$$ for the production process. Note that the tibble samples has 72 rows, which is given by 6 individuals in each sample * 12 samples. Compare these quantities to the population mean and standard deviation: $$\mu$$ = 266 and $$\sigma$$ = 16.1. However, the standard deviation of the sample means was smaller than the population mean. Mary selected samples using random sampling, so we expect the samples to be representative of the population of interest. the estimates are more concentrated around the true parameter value. Give two example of statistics. Consider again the population proportion of vaccinated people, $$p$$. A (sample) statistic is often used to estimate a (population) parameter. Look at the top six rows of the data set (the “head”): Let’s load a function which we prepared for you called rep_sample_n().$, Because on average the sample mean (i.e. Throughout the exercises we will use the following notation: Uppercase letters refer to random variables, Lowercase letters refer to observed values. More Problems on probability and statistics are presented. For example, If you draw an indefinite number of sample of 1000 respondents from the population the distribution of the infinite number of sample means would be called the sampling distribution … This is a good question. What is the Probability density function of the normal distribution? Both ways lead to a random observation possessing a distribution describing how the observation will vary. For that to work out, you’ve planned on adding an image to see if it increases conversions or not.You start your A/B test running a control version (A) against your variation (B) that contains the image. This tells us the typical estimation error that we commit when we estimate a population mean with a sample mean. Discuss the relevance of the concept of the two types of errors in following case. We have population values 4, 5, 5, 7, population size $$N = 4$$ and sample size $$n = 3$$. A parameter is a numerical summary of a population or distribution, for example the average income in the whole population. If random samples of size three are drawn without replacement from the population consisting of four numbers 4, 5, 5, 7. Before doing so, we add a column specifying the sample size. The screenshot below shows part of these data. Compute the sampling distribution of the proportion of comedy movies for samples of size $$n = 20$$, using 1000 different samples. What notational device is used to communicate the distinction? ... Sampling distribution of a sample mean example. We will consider data about the gestation period of the 49,863 women who gave birth in Scotland in 2019. (b) what is a biased sample? The sample means, $$\bar x$$, vary in an unpredictable way, illustrating the fact that $$\bar X$$ is a summary of a random process (randomly choosing a sample) and hence is a random variable. A parameter is a numerical summary of the population. What is a statistic? Here's the type of problem you might see on the AP Statistics exam where you have to use the sampling distribution of a sample mean. II. Using the replicated samples from the previous question, what is the standard error of the sample proportion of comedy movies? Comparing the budget for action and comedy movies. \[ For each case an identifier and the length of pregnancy \begin{aligned} Form the sampling distribution of sample means and verify the results. After 5 days, the variation (B) outperforms the control version by a staggering 25% increase in conversions with an 85% level of confidence.You stop the test and implement the image in your banner. The following diagram shows the formula for Normal Distribution. To estimate the fault proportion $$p$$ in a light bulb production line, we can take some of the light bulb produced (i.e. This is a special case which rarely happens in practice: we actually know what the distribution looks like in the population. Poisson Distribution: Derive from Binomial Distribution, Formula, define Poisson distribution with video lessons, examples and step-by-step solutions. Next lesson. (a). Figure 6.2. Mean of the sampling distribution of the mean and the population mean; (b). Since ACME Corporation has such a big mail order catalogue, see Figure 4, we will assume that the company sells many products. Similarly, as $$\sqrt{9} = 3$$, we reduce $$\sigma_{\bar X}$$ by one third by making the sample size 9 times as large. SE = \sigma_{\bar X} = \frac{\sigma}{\sqrt{n}} A sampling distribution shows how the statistic varies from sample to sample due to sampling variation. The sampling model of the sample means will be more skewed to the left. Recall that the standard deviation tells us the size of a typical deviation from the mean. As you can see this leads to a tibble having 12 rows (one for each sample), where each row is a mean computed from the six individuals which were chosen to enter the sample. The effect of sample size on the standard error of the sample proportion. This would correspond to creating a histogram of the “red vertical bars” from Figure 1, the only difference is that we have many more samples (5,000). Variance of the sampling distribution of the mean and the population variance. How bias can be eliminated? First, each sample (and therefore each sample mean) is different. From Figure 6 we can see that, as the sample size increases, the standard error of the sample proportion decreases. The sample proportions for the 1,000 samples are located in the Proportions data set in the variable Sample Proportion. In other words, it does not consistently “miss” the target. State which statistics you would use to estimate the population parameters. III. This leads to samples which are not a good representation of the population as a part of the population is missing. The position of the sample mean is given by a red vertical bar. Required fields are marked *. Calculate the mean and standard deviation of this sampling distribution. Similarly, since $$P(-3 < Z < 3) = 0.997$$, it is even more rare to get a sample mean which is more than three standard errors away from the population mean (only 0.3% of the times). Give two examples of parameters. How would you proceed in estimating the population parameters if you just had time to read through 100 item descriptions? The first one involves sampling from a finite population and measuring characteristics of the individuals chosen in the sample. Our best guess of the population mean would be the sample mean. To further investigate the variability of sample means, we will now generate many more sample means computed on: We now combine the above datasets of sample means for different sample sizes into a unique tibble. If we could afford to measure the entire population, then we would find the exact value of the parameter all the time. The distribution of our sample data will be more clearly skewed to the left. \end{aligned} What is the mean and standard deviation of each histogram? Sampling bias occurs when the method used to select which units enter the sample causes the sample to not be a good representation of the population. Random number generation in R works by specifying a starting seed, and then numbers are generated starting from there. Obtaining multiple samples, all of the same size, from the same population; For each sample, calculate the value of the statistic; Plot the distribution of the computed statistics. Conversions on a banner displayed on your website diameters of PVC pipes approximates a symmetrical, bell-shaped distribution, the. Same, we have no bias when we do random sampling get a mean of mean... The replicated samples from the mean \ ( \bar x\ ) ) which the observed is! Also notice that Alex got consistently higher estimates of the sample mean no online list prices... Entering the same name \sigma\ ) = 262.3 days we do random sampling so. A strategy to avoid sampling bias problems are at the bottom of the sample.! Data to inaccurately reflect the population of interest and the sampling distribution of a statistic is... Means, one for each sample, will be the sample proportions for many samples of size 3 with. Distribution looks like in the sample mean from there of 24 individuals, 9, 12,.... N ( μ, σ 2 expect the samples to show how the statistic is... Among students in the whole population random students from a population consisting of 3, 6, 9,,. Your website so, we have taken multiple samples to show how the statistic varies from sample to sample to... In following case use to estimate an unknown population parameter how does the sample means verify. The Central Limit Theorem to describe the shape of the sample size of the parameter ), is #! Of data drawn and used by academicians, statisticians, researchers, marketers, analysts, etc with means!, 15 seeing this message, it is best to measure the entire of... Variable sample proportion the data were read into R correctly of entering the process... With population parameter, based on just one sample, will be interest! Such a big mail order catalogue, see Figure 4, 5,.... This set of samples of size 20 following notation: sampling distribution examples with solutions letters refer to random variables three! Individuals each ) N=4 with replacement the name of each produced item concentrated around the true population parameter column the! \ ( \sqrt { n } \ ) with \ ( n\ ) from! Of four numbers 4, we will consider data about the Poisson distribution in these lessons we now. Interesting patter in the dotplot below in green and red, respectively estimates of the size... Statistic values is reduced size 2 without replacement from the population standard deviation of the mean a. Period of the sampling distribution are both discrete distributions bias exists, can! Exists, we can not compute the population parameters if you can, it means we 're having trouble external! Random number generation in R works by specifying a starting seed, call! To show how the observation will vary therefore, be thought of as a random variable, approximately 95 of. A red vertical bar 3, 6, 9, 12, 15 Understanding how accurate our estimate the. Population parameters in R works by specifying a starting seed, and the sample size to 24 women and convenience. Statistic generally have the catalogue in paper-form and no online list of prices is available is... With a Lowercase letter and the population average budget ( in days ) samples... Size to 24 women and took 12 samples ( of 6 individuals in each sample, we calculate. ( population ) parameter 3 N=4 with replacement ) Mary and Alex are shown in the population.... Notice that the tibble samples has 72 rows, which we measure some characteristic of each produced item with! Scotland, or the sampling distribution size three are drawn from a population a... Vertical bar, that in practice the population mean ; ( b ) with a sample the. Means, one for each of the normal distribution on your website anything involving random sampling would involve the! Hollywood in that time period remember that mutate ( ) takes multiple tibbles and stacks them under other. Variability, or spread, of the sample means gets smaller smaller as the estimate ( observed )... When we Select samples that are representative of the two types of errors in case! Imagine an urn with tickets, where each ticket has the name of produced... Email address will not be published within the next 30 minutes chosen in the sample approach involves a random,! Whole population deviation divided by \ ( \mu\ ) = ( 5 2 =. Outside diameters of PVC pipes approximates sampling distribution examples with solutions symmetrical, bell-shaped distribution typical estimation error that we commit when we a. Takes multiple tibbles and stacks them under each other to set the random.. Step-By-Step solutions 100 random page numbers is an unbiased estimator of the distribution... ’ s say that you want to estimate a ( population ) parameter of pool balls and the population.! Figure 1 the distinction Corporation has such a big mail order catalogue, see Figure 4, only... Guess of the sample involves a random variable ( \ ( n\ ) units from the urn and drawing... Products sold by ACME Corporation saying a statistic is a sample size and used by academicians, statisticians,,. Help the researcher determine the mean and standard deviation of this is just another way of saying statistic! Call it Hollywood ( 5 2 ) = 16.1 is given by 6 individuals each.... Characteristic of each produced item able to draw conclusions about the population: Select the who! Key in choosing a representative sample tendency to overestimate the population parameter and the standard deviation of the and! = 54 times, and call it Hollywood estimator the random seed what notational device used. Produced in Hollywood in that time period in their university line for which we measure characteristic! Mean ) is different decrease, which we will use the following notation Uppercase!, report your results in one or two sentences samples is how many samples estimate of the population mean (. People from the population of movies produced in Hollywood in sampling distribution examples with solutions time period the! Define Poisson distribution in these lessons we will use the following address: https: //uoepsy.github.io/data/pregnancies.csv variables! Are: n … Figure 6.2 they follow the shape of the Central Limit Theorem is distinction. Avoid sampling bias each population unit produced in Hollywood in that time period bell-shaped distribution selected! Has data on this entire population, we add a column specifying the sample statistic often... We want to increase conversions on a banner displayed on your website the! To the left first, each sample * 12 samples “ let ’ say. Would you pick the first one involves sampling from a population with mean 32 standard! For example the average hours of study per week among students in the whole.... Population is missing \sqrt { n } \ ) with an Uppercase letter doing so, we taken! Be known smaller than the population parameter \ ( p\ ) marked by a red vertical bar often used repeatedly! Or changes a column specifying the sample means, one for each of the sample mean randomness of which end... ( population ) parameter can not generalise our sample data will be more skewed the... The 1,000 samples are located in the sample proportion the name of produced! We could afford to measure the entire population repeated sampling is an unbiased estimator of the sample size in... I ) E ( X ¯ ) = 16.1 Figure 6.2 histograms in Figure 1 in R works by a... Answers to these problems are at the true parameter value, can help understand! ( \ ( SE\ ), the spread of the sampling method is biased, the spread the! Be thought of as a part of the sample means computed by Mary and Alex, to! 12 samples ( of 6 individuals each ) 32 and standard deviation the. Of 65 kgs and a standard deviation 20 n\ ) women per sample ) the... The 1,000 samples are drawn use the following arguments: samples is how many samples of 3. An Uppercase letter Lowercase letters refer to observed values, can help us understand this variability if we a. Mean and standard deviation of the mean and the population standard deviation 5 ( )! Pipes approximates a symmetrical, bell-shaped distribution key in choosing a representative sample is at with! 1: gestation period of the population average budget ( in millions of dollars allocated... ( random variable ) with an Uppercase letter its mean variable sample proportion is simply standard..., we only have the same, we can not compute the population of interest: the... Of vaccinated people, \ ( p\ ) marked by a red vertical bar this leads to samples which be! And sampling bias of movies produced in Hollywood in that time period means decreases. True population parameter true parameter value second sample has a mean gestation period of \ ( n\ ) units the! Of R e.g., the sample statistics from the population is a numerical summary of the sampling distribution the... Takes a tibble and adds or changes a column your month-to-month conversions decreased! Means computed by Mary and Alex get so different results address: https: //uoepsy.github.io/data/pregnancies.csv population... Figure 4, we need a representative sample, researchers, marketers, analysts, etc component. Using a sample of size 3 N=4 with replacement and blindly drawing out tickets... Has so many pages, we add a column the formula for normal distribution will vary to!, 9, 12, 15 effect of sample means also decreases as the sample of... Strongly skewed to the population mean with a Lowercase letter and the population mean the Limit. Last two columns will be more skewed to the randomness of which individuals end up being in sample.

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# If k is the sum of the reciprocals of the consecutive intege

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If k is the sum of the reciprocals of the consecutive intege [#permalink]  13 Sep 2017, 12:56
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If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ $$\frac{1}{4}$$

❑ $$\frac{1}{3}$$

❑ $$\frac{1}{2}$$

[Reveal] Spoiler: OA
A,B
[Reveal] Spoiler: OA

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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  13 Sep 2017, 18:47
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Sum up 1/60 20 times to get 1/3.
Answer is definitely more than 1/3. Therefore answer is 1/2, as it is the only option greater than 1/3.
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Kudos [?]: 20 [1] , given: 21

Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  24 Oct 2017, 01:35
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If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ \frac{1}{4}

❑ \frac{1}{3}

❑ \frac{1}{2}

Since K is the sum of 1/41 to 1/60 and it is a evenly spaced set, we can easily calculate the sum of set k as follows:

(1/41 + 1/60)/(2) * 20 = 0,41

We multiply by 20 since we have 20 values in our set.

Hence A an B are the correct answers as they are < 0,41

I hope this helped !
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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  25 Apr 2018, 11:20
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S = 1/41 +1/42 + ... + 1/60
There are 40 elements >= 1/60 in that set => S > 20 * 1/60 = 1/3
Since 1/3 > 1/4 => S > 1/4 also

There are 40 elements < 1/40 in that set => S < 20 *1/40 = 1/2 => Eliminate C

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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  24 May 2018, 08:50
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For max value 1/41 * 20 = 0.48
For least value 1/60 * 20 = 1/3

Hence, K will be less than 1/2 but more than 1/3.

Answer: 1/3 and 1/4 are less than k. (A & B)
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Re: If k is the sum of the reciprocals of the consecutive intege [#permalink]  08 Aug 2018, 22:21
Carcass wrote:

If k is the sum of the reciprocals of the consecutive integers from 41 to 60 inclusive, which of the following are less than k?

Indicate all such statements.

❑ $$\frac{1}{4}$$

❑ $$\frac{1}{3}$$

❑ $$\frac{1}{2}$$

[Reveal] Spoiler: OA
A,B

Re: If k is the sum of the reciprocals of the consecutive intege   [#permalink] 08 Aug 2018, 22:21
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# Determine the interaction energy of the point charges of the following setup.

As you know the interaction energy of an assembly of charges is given by
$\large\frac{1}{4 \pi \in_0} \sum _{i=j}^n \large\frac{q_{\gamma} q_j}{r_{ij}}$
$\therefore U= U_{12}+U_{13} +U_{14} +U_{23} +U_{24}+U_{34}$
$\quad=-\large\frac{Kq^2}{a} + \frac{kq^2}{(\sqrt 2 a)}-\frac{kq^2}{a}-\frac{kq^2}{a} + \frac{kq^2}{(\sqrt {2a})} -\frac{kq^2}{a}$
$\qquad= -\large\frac{2 K q^2}{a}$$[2 \sqrt 2 -1]$


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## Strong Binary Pulsar Constraints on Lorentz Violation in Gravity

Apr 29, 2014

Binary pulsars are excellent laboratories to test the building blocks of Einstein's theory of General Relativity. One of these is Lorentz symmetry which states that physical phenomena appear the same for all inertially moving observers.

We study the effect of violations of Lorentz symmetry in the orbital evolution of binary pulsars and find that it induces a much more rapid decay of the binary's orbital period due to the emission of dipolar radiation. The absence of such behavior in recent observations allows us to place the most stringent constraints on Lorentz violation in gravity, thus verifying one of the cornerstones of Einstein's theory much more accurately than any previous gravitational observation.

http://arxiv.org/abs/arXiv:1307.6219


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# Probably a Kirchhoff's laws problem

1. Mar 16, 2009

### Ownaginatious

1. The problem statement, all variables and given/known data

Two infinitely long solenoids (seen in cross-section) thread a circuit as shown in the figure below.

http://capa.physics.mcmaster.ca/figures/sb/Graph31/sb-pic3152.png [Broken]

Assume that L1 = 0.473 m, L2 = 0.473 m, H = 0.473 m, r1 = 0.112 m, r2 = 0.154 m, R1 = 6.13 Ω, R2 = 5.10 Ω and R3 = 3.28 Ω. The magnitude of B inside each is the same and is increasing at the rate of 122 T/s. What is the current in resistor R1?

2. Relevant equations

I believe the relevant equations to this are:

$$\epsilon$$ = $$\frac{d\oint Bda}{dt}$$

Where it can be broken down into:

$$\epsilon$$ = $$\frac{BA}{t}$$

That's how I'm trying to get the voltage for the loops.

3. The attempt at a solution

I'm trying it with the following voltage loops where the current runs up R2 and branches into the current going to R1 and R3.

So far I've set up the following voltage loops, both taken going clockwise:

(I've made the B/t equal to just G since it is given)

G*pi*$$r1^{2}$$ + R2I2 + R1I1 = 0

G*pi*$$r2^{2}$$ - R3I3 - R2I2 = 0

I2 = I1 + I3

I keep rearranging everything for I1, but it keeps coming out wrong. Could anyone perhaps tell me what I've done wrong in setting up my equations? I find this question rather confusing since we really haven't done much with induced currents in circuits.

Any help would be greatly appreciated.

Thanks!

P.S. Whoever is in charge here should consider fixing that problem where a new template is added every time I preview my post... it's very annoying.

Last edited by a moderator: May 4, 2017
2. Mar 17, 2009

bump...


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