Week 9: Assumptions of Multivariable Linear Regression

Question 1

How do you check the distribution of continuous variables?

Answer 1

<hist varname, freq normal> to create a histogram and overlay a normal distribution

Question 2

How do we call a coefficient when the dependent variable decreases as the independent variable increases?

Answer 2

Negative coefficient

Question 3

What does the coefficient represent?

Answer 3

The change in the dependent variable for a one-unit change in the predictor, holding other variables constant

Question 4

What is the significance of residuals in regression analysis?

Answer 4

Residuals measure the difference between observed and predicted values, indicating model fit

Question 5

How can you test if residuals are normally distributed using a kernel density plot?

Answer 5

<kdensity resid_varname, normal> and overlay a normal curve to check for alignment

Question 6

How does a pnorm plot help in assessing normality of residuals?

Answer 6

It compares the cumulative distribution of residuals to a normal distribution; closer alignment suggests normality. Qnorm plots show deviations from normality in the middle range of data

Question 7

What does deviation from the line in a qnorm plot represent?

Answer 7

Deviation at the tails indicates non-normality, suggesting potential outliers or skewness. Shows deviations from normality at the extremities.

Question 8

Why is normality of residuals important in linear regression?

Answer 8

Normal residuals ensure valid hypothesis testing and confidence intervals

Question 9

What is the purpose of a residual vs fitted plot?

Answer 9

It checks for patterns that indicate violations of linearity, equal variance, or non-normality

Question 10

How can you assess if the linearity and equal variance assumptions are met?

Answer 10

Look for random scatter in a residual vs fitted plot; fanning or patterns suggest heteroscedasticity
In a residual vs fitted plot, non-linearity is shown by a pattern, whereas unequal variance is shown by a funnel shape

Question 11

What does the presence of leverage points indicate?

Answer 11

Leverage points are influential observations that can disproportionately affect model fit

Question 12

How can you identify leverage points in regression analysis?

Answer 12

Plot residuals or fitted values against predictors and look for isolated points

Question 13

What does multicollinearity indicate in a regression model?

Answer 13

High correlation between predictors can distort coefficients, making them unreliable

Question 14

How can multicollinearity be detected?

Answer 14

Calculate correlation coefficients between predictors; values near +/- 1 indicate multicollinearity. Use command <cor></cor>

Question 15

Why might adding two highly correlated predictors distort regression results?

Answer 15

The shared variance between predictors reduces the model’s ability to isolate individual effects

Question 16

What does recoding or transforming a variable achieve in regression analysis?

Answer 16

It can improve the fit by correcting for skewness, non-linearity, or address data inaccuracies (e.g., age grouping)

Question 17

Why is it important to check for missing data after transformations?

Answer 17

Transformations can exclude cases, reducing sample size and potentially biasing results

Question 18

How can you improve the normality of residuals?

Answer 18

Apply transformations (e.g., log, square root) to the dependent variable to reduce skewness

Question 19

What does a funnel shape in a residual plot suggest?

Answer 19

It indicates heteroscedasticity - variance of residuals increases with fitted values

Question 20

What is the implication of non-linearity in residual plots?

Answer 20

Non-linearity suggests that the relationship between predictors and outcome may not be adequately captured by the model

Question 21

Why might adding interaction terms improve model fit?

Answer 21

Interactions account for cases where the effect of one predictor depends on the level of another predictor

Question 22

What does a constant (_cons) represent?

Answer 22

It is the predicted value of the dependent variable when all predictors are zero

Question 23

How can transformations improve model assumptions?

Answer 23

They can stabilise variance, reduce skewness, and make relationships more linear

Question 24

What does a high peak in a histogram indicate about the distribution?

Answer 24

It may suggest a large concentration of data around a specific value, potentially indicating skewness or rounding

Question 25

Why is assessing the distribution of predictors and outcomes crucial in regression?

Answer 25

Non-normality or outliers can lead to biased estimates and affect the validity of the model

Question 26

How does excluding extreme observations affect model fit?

Answer 26

It can reduce leverage effects and improve stability, but may also remove meaningful data

Question 27

What is the purpose of fitting multiple models with different predictors?

Answer 27

It helps to identify the best combination of variables and assess the robustness of results

Question 28

Why is it useful to visualise residuals after each regression model?

Answer 28

It allows for continuous assessment of model assumptions and fit

Question 29

How do you generate residuals for a regression model?

Answer 29

<predict new_varname, resid>
This will generate a new variable. This cannot be overwritten by any other residuals variable created - subsequent variables must have a new name

Question 30

How do you plot a pnorm and qnorm plot to examine the normality of residuals?

Answer 30

<pnorm>
<qnorm>
</qnorm></pnorm>

Question 31

What command do you use to check normality of residuals using a residual vs fitted value plot?

Answer 31

<rvfplot, yline(0) msymbol (Oh) msize(tiny)>
The <yline> option draws a line at 0 where the residuals should be densest
The <msymbol> option is used to change the default symbol which is now set at a hollow circle</msymbol></yline>

<msize> changes the size of the symbol
</msize>

Question 32

How can we make a variable containing the fitted values and why would we want to do this?

Answer 32

<predict fit_varname if e(sample)>
The fitted values can be plotted on a scatterplot to see where there may be issues with fitted values (may be far away from the others) -
<scatter fit_varname varname condition, msymbol (Oh) msize(tiny)> Or
<twoway (scatter fit_varname varname condition, sort) (scatter fit_varname varname condition, sort)>

Question 33

How can you check the options for transforming a variable?

Answer 33

<gladder>
</gladder>

Question 33

How do you perform a log transformation on a variable?

Answer 33

<gen log_varname=log(varname)>