Linear Regression

In the babies.dta full dataset, generate a covariate called painind defined as 1 if the infant experienced severe pain upon receiving the shot (pain0 = 7) and as 0 otherwise. In Stata, you can use the commands:

generate painind = 0
replace painind = 1 if pain0 == 7

Fit a linear regression model with total cry time as the outcome; and with group and painind (the severe pain indicator) as covariates. The regression model is:


where .

1. Using the notation from the model above, what are your estimates of the regression coefficients and residual standard deviation?

2. Using the fitted regression model, estimate the average change in cry time for infants with severe pain versus those without severe pain, holding group constant. Provide a 95% confidence interval for this estimate.

Estimate:

unanswered


95% Confidence interval Lower Bound:

unanswered


95% Confidence interval Upper Bound:

42.38

0.0035 - 0.045

To answer these questions in Stata, you can follow these steps:

1. Generating the covariate called "painind":
- Open Stata and load the "babies.dta" dataset.
- Type the following command: `generate painind = 0` to create the variable "painind" with initial values of 0 for all observations.
- Then, use the command `replace painind = 1 if pain0 == 7` to set the value to 1 for those infants who experienced severe pain upon receiving the shot (pain0 = 7).

2. Fitting the linear regression model:
- To fit the linear regression model with total cry time as the outcome and with "group" and "painind" as covariates, use the following command:
`regress crytime group painind`

Now, let's answer the questions using the notation provided in the model:

1. Estimates of the regression coefficients and residual standard deviation:
- After running the regression, you will see the estimates output. Look for the coefficients of "group" and "painind".
- The estimate of the regression coefficient for "group" (denoted as β_1 in the model) represents the average difference in cry time between the groups, adjusting for painind.
- The estimate of the regression coefficient for "painind" (denoted as β_2 in the model) represents the average difference in cry time between infants with severe pain (painind = 1) and those without severe pain (painind = 0), holding group constant.
- The residual standard deviation estimate (denoted as σ in the model), also known as the standard error of the regression or the root mean square error, measures the average distance between the observed cry times and the predicted values from the regression model. It quantifies the variability of the cry times that is not explained by the covariates.

2. Estimating the average change in cry time for infants with severe pain versus those without severe pain, holding group constant:
- To estimate this average change, we can look at the estimate for the coefficient of "painind" (β_2 in the model).
- The estimate represents the expected change in the cry time for infants with severe pain compared to those without severe pain, adjusting for the effect of "group".
- To obtain the 95% confidence interval for this estimate, we need to look at the confidence interval associated with the coefficient of "painind" in the regression output.
- The lower bound of the confidence interval represents the lower limit of the range of values that we are 95% confident contains the true average change.
- The upper bound of the confidence interval represents the upper limit of the range of values that we are 95% confident contains the true average change.

The specific values for the estimates and confidence intervals cannot be determined without running the regression on the actual dataset.