1. Covariate adjustment in experimental design
Let's now explore covariates in experimental design and analysis, and how they can be used to minimize confounding. We'll also learn about ANCOVA, or analysis of covariance, for evaluating treatment effects while controlling for covariates.
2. Introduction to covariates
Recall that covariates are variables that are not of primary interest but are related to the outcome variable and can influence its analysis.
Including covariates in statistical analyses is crucial for reducing confounding, which occurs when an external variable influences both the dependent variable and independent variable(s). By adjusting for covariates, researchers can isolate the effect of the independent variable on the outcome, minimizing the influence of confounders.
Accounting for covariates in experimental design and analysis controls for variability that is not attributable to the primary variables being studied. This leads to more valid conclusions about the relationship between the independent and dependent variables, as the analysis better reflects the true effect by isolating it from the influence of covariates.
Consider the investigation of a new teaching method's effectiveness on student test scores. Here, the primary variables of interest are the teaching method (independent variable) and the student test scores (dependent variable). However, students' prior subject knowledge serves as a crucial covariate because prior knowledge can significantly impact learning outcomes, yet it's not the main focus of the study.
3. Experimental data example
Let's bring back our plant growth data and set it to experimental data as the exp_data DataFrame, keeping Fertilizer_Type as treatment and Growth_cm as response.
4. Covariate data example
The covariate_data DataFrame also includes Plant_ID identifiers for each subject, again ranging from 1 to 120, ensuring each subject's covariate data is matched with their experimental data. Watering_Days_Per_Week is another variable measured for each plant. Recall that covariates are additional variables potentially influencing the outcome and are included in analyses to control for their effects.
5. Combining experimental data with covariates
Combining the experimental with covariate data is a crucial step in adjusting for covariates. We use pandas' merge function to combine DataFrames; we do this on the Plant_ID to ensure each that subject's experimental and covariate data are aligned.
6. Adjusting for covariates
To adjust for covariates in our analysis, we employ ANCOVA, or analysis of covariance, using the ols model from statsmodels.
This ols() function takes a formula that specifies the dependent and independent variables.
Growth_cm is the dependent variable we're interested in, which we want to model using the Fertilizer_Type, the categorical independent variable representing different groups in the experiment, and the potential covariate, Watering_Days_Per_Week, to control for its effects.
The first portion of summary output provides details on the significance of the model; it show a large p-value here of 0.531, which implies a lack of support for covariates affecting the model.
7. Further exploring ANCOVA results
Looking at the second and third rows of this second portion of output from summary, we see that the factors and covariate each have large p-values of 0.760 and 0.275, concluding that each of them alone are not significant predictors of growth for this model.
8. Visualizing treatment effects with covariate adjustment
This seaborn lmplot shows treatment effects adjusted for the covariate. The regression lines for each treatment category offer a visual representation of how treatment effects trend across different levels of the covariate. We see that Organic remains relatively constant going from 1 watering to 7 Watering_Days_Per_Week. Synthetic shows an increase. The crossing regression lines suggest we may want to add an interaction term of Watering_Days_Per_Week by Fertilizer_Type in another model. Parallel lines would suggest a lack of interaction.
9. Let's practice!
Armed with an understanding of covariate adjustment, you're now ready to put this knowledge into practice.