--- title: "Observational Studies and Stratification" subtitle: | | Additional Topic | Math 219 author: "Yurk" format: revealjs: theme: beige editor: source --- ```{r} #| include: false #| echo: false library(tidyverse) library(infer) library(openintro) library(broom) library(HH) ``` ## Reducing Unexplained Variability - The primary role of blocking in experiments is to reduce the unexplained variability in the response - **Stratifying** plays a similar role in observational studies - Groups of observational units are formed with similar values of the **stratifying variable** - Stratifying variable is accounted for as a source of variability in the analysis (similar to blocking in experiment) ## Confounding Variables - **Confounding variables** are associated with both the dependent variable and the independent variable - Confounders can obscure the true relationship between the variables of interest - In experiments, randomization ensures that treatment groups are similar in terms of confounding variables - This is not the case in observational studies - Often, the stratifying variable is a potential confounding variable ## Smoking and birth weight ```{r} #| include: FALSE #| echo: FALSE data(births14) births14 <- births14 |> drop_na(habit, weight, weeks, premie) ``` - Do infants whose mothers smoke have a different mean birth weight than infants whose mothers do not smoke? - `births14` [^1] dataset - Random sample of 1,000 cases from US birth data set from 2014 (19 removed with missing values) - `habit` is smoking habit ("smoker" or "nonsmoker") - `weight` is birth weight in pounds - `premie` birth is premature (premie) or full-term [^1]: `births14` is from the `openintro` package --- - First, we will include smoking habit as a single explanatory variable (no stratifying variable) - Although it would be appropriate to compare means for smoking and non-smoking mothers using a t-test (see Ch 20 slides), an F-test gives us the same results ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ habit, data = births14) |> anova() |> tidy() ``` ## `premie` as a Stratifying Variable - The length of the pregnancy is expected to explain some of the variation in birth weight - It also has the potential to be a confounding variable if there is an association between smoking and length of pregnancy - We will use `premie` as a stratifying variable in our analysis ## Statistical Model with a Stratifying Variable Statistical model for the $k$th observation in group $i$ and stratum $j$, $$y_{ijk}=\mu+\alpha_i+\beta_j+\varepsilon_{ijk}$$ - $\mu$ is the overall mean - $\alpha_i$ is the differential effect of group $i$ - $\beta_j$ is the differential effect of stratum $j$ - $\varepsilon_{ijk}\sim N(0,\sigma^2)$ is the noise ## ANOVA Table - $SS_{premie}$ accounts for the effect of `premie` - $SS_{habit}$ accounts for both variables, but then $SS_{premie}$ is subtracted off ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ premie + habit, data = births14) |> anova() |> tidy() ``` --- - Interestingly, $p$-value for `habit` is higher when accounting for `premie` in the model - This is the opposite of what we would expect to see in an experiment, and is due to association between `premie` and `habit` ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ premie + habit, data = births14) |> anova() |> tidy() ``` --- We can also see the effects of this association if we compare the coefficient of the corresponding linear models. ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ habit, data = births14) |> tidy() ``` `habit` has a smaller effect when we account for `premie` ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ premie + habit, data = births14) |> tidy() ``` ## Association between `habit` and `premie` ```{r} #| include: TRUE #| echo: FALSE births14 |> ggplot(aes(habit)) + geom_bar(aes(fill = premie), position = position_fill()) + labs(y = NULL) + theme_minimal() ``` --- - Due to the association between `habit` and `premie` it is difficult to fully separate their effects on birth weight - E.g., whether the mother smokes or not *could* affect whether the baby is a premie or not which could impact weight - $SS_{habit}$ measures the variation in the response that is attributed to `habit` but does not include the variation that is jointly attributed to the two explanatory variables --- - If we reverse the order of the variables, we can also measure the variation that is attributed to `premie` without variation that is jointly attributed to the two variables ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ habit + premie, data = births14) |> anova() |> tidy() ``` ## ANCOVA - In the previous analysis we used `premie` as a stratifying variable - The dataset also includes the variabe `weeks`, the length of the pregnancy in weeks - We can conduct an alternative analysis using `weeks` as a (numeric) covariate instead of stratifying by `premie` - Statistical model (parallel lines) for the $j$th observation in the $i$th group: $$y_{ij}=\mu + \alpha_i + \beta(X_{ij} - \bar{\bar{X}})+\varepsilon_{ij}$$ --- - `weeks` as a covariate explains more of the variation in the response than `premie` ($SS_{weeks} = 483$ vs. $SS_{premie}=374$) - The resulting model explains more of the variability in the response than the model that included `premie`, so $SSE$ is smaller - As a result, the $F$ statistic is larger, and the $p$-value is smaller ```{r} #| include: TRUE #| echo: TRUE lm(weight ~ weeks + habit, data = births14) |> anova() |> tidy() ``` ## Conclusions - Both analyses lead us to conclude that there is a significant association between smoking and birth weight when we also account for the length of pregnancies - We reach the same conclusion whether we treat the length of pregnancy as a categorical variable (`premie`) or a numeric variable (`weeks`) - Because this is an observational study, we cannot conclude that smoking causes a difference in birth weight - It is possible that there are other confounding variables that we have not accounted for in the model