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• 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?

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• 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:

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95% Confidence interval Lower Bound:

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95% Confidence interval Upper Bound:

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• 3. Again, use the notation above for the regression model. The correct interpretation for is:

Infants in the intervention group have times the risk of experiencing an increase in cry time compared to infants in the control group Infants in the intervention group have times the risk of experiencing an increase in cry time compared to infants in the control group after controlling for pain experienced by the infant Infants in the intervention group on average have change in cry time compared to the control group. Infants in the intervention group on average have change in cry time compared to the control group, after controlling for severity of pain experienced by the infant upon receiving the shot.

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• 4. Using the regression model, estimate the average cry time in the following groups:

Control group infants with severe pain upon receiving the shot

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Control group infants without severe pain upon receiving the shot

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Intervention group infants with severe pain upon receiving the shot

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Intervention group infants without severe pain upon receiving the shot

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• 5. Without using the regression model, estimate the mean cry time in the following groups:

Control group infants with severe pain upon receiving the shot

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Control group infants without severe pain upon receiving the shot

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Intervention group infants with severe pain upon receiving the shot

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Intervention group infants without severe pain upon receiving the shot

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We refer to these as 'non-parametric' estimates, because they do not rely on modeling assumptions (whereas the estimates in question 4 are based on the linear regression model).

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• 6. Compare your estimates of the group-specific means from the regression model to the "non-parametric" estimates above. In large sample sizes, would you expect the "non-parametric" estimates or the regression based estimates to have less bias (e.g. be closer to the true group-specific means in the population)?

non-parametric regression

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• 7. With continuous covariates, we cannot estimate the means using the non-parametric method as above due to the "curse of dimensionality." This is because:

some continuous variables have skewed distributions there is typically only one observation per continuous variable hence the mean cannot be estimated well

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• 8. Suppose that sex is an effect modifier of the association between group and cry time. Which of the following is a correct way to analyze the data?

Construct a linear regression model with sex as a covariate. Construct two separate linear regression models: one among male infants and one among female infants. Construct a linear regression model, but do not control for sex as a covariate because this is a randomized clinical trial.

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