• 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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1. 29.7833
-7.679168
12.61215
21.766
2. 12.612
5.236
19.989
please help me with cohort study
Thanks a ton for your reply. Can you answer 3,4 and 5 please
To answer the questions about linear regression, you need to perform the following steps:
1. Generate the covariate called painind:
- Use the Stata command `generate painind = 0` to create a new variable named `painind` with an initial value of 0.
- Then, use the command `replace painind = 1 if pain0 == 7` to assign a value of 1 to `painind` if the variable `pain0` is equal to 7.
2. Fit the linear regression model:
- Use the linear regression command in Stata with the outcome variable `total cry time` and the covariates `group` and `painind`.
Once you have fit the regression model, you can answer the questions:
1. To obtain the estimates of the regression coefficients and residual standard deviation, you can use the Stata command `regress`. The coefficients will be provided in the output, along with the residual standard deviation.
2. To estimate the average change in cry time for infants with severe pain compared to those without severe pain, holding group constant, you can use the coefficients from the linear regression model. The coefficient for the covariate `painind` represents the average change in cry time associated with having severe pain compared to not having severe pain. You can calculate the confidence interval using the standard error of this coefficient.
3. The interpretation of the coefficient `beta1` is that infants in the intervention group have a `beta1` times the risk of experiencing an increase in cry time compared to infants in the control group after controlling for the severity of pain experienced by the infant upon receiving the shot.
4. To estimate the average cry time in different groups, you can use the regression model to obtain the predicted cry time for each group. Replace the `group` variable with the relevant group and the `painind` variable with the relevant severe pain indicator (0 for without severe pain, 1 for with severe pain). Then, use the regression equation to calculate the predicted cry time.
5. To estimate the mean cry time in different groups without using the regression model, you can calculate the mean cry time directly from the data. Calculate the mean cry time separately for each group and each combination of severe pain indicator and group.
6. In large sample sizes, the regression-based estimates are expected to have less bias and be closer to the true group-specific means in the population compared to the non-parametric estimates. Regression models can account for potential confounding variables and provide more accurate estimates when the assumptions of the model are met.
7. The "curse of dimensionality" refers to the difficulty of estimating means using non-parametric methods when there are continuous covariates. The non-parametric methods usually rely on grouping or categorizing the continuous variables, which can lead to biased estimates. Since continuous variables can have an infinite number of values, it becomes challenging to estimate the means accurately.
8. To analyze the data when sex is an effect modifier of the association between group and cry time, you can construct two separate linear regression models: one for male infants and one for female infants. This allows you to examine the association between group and cry time separately for each sex, taking into account the effect modification.