Week 6: Linear Regression and Correlation

Question 1

Which commands can be used to explore variables?

Answer 1

<codebook>
<histogram>
<summarize varnames, detail>
</histogram></codebook>

Question 2

What are the independent and dependent variables in the association between age and executive function?

Answer 2

Age (independent)
Executive function (dependent)

Question 3

What command is used for linear regression?

Answer 3

<regress>
</regress>

Question 4

What does the intercept (β0) represent in regression analysis?

Answer 4

The predicted value of the dependent variable when the independent variable is 0

Question 5

What is the equation for simple linear regression?

Answer 5

y = β0 + β1x

Question 6

How do you centre a variable around the mean?

Answer 6

<sum varname, meanonly>
<gen varname_cent = varname - r(mean)>

Question 7

How is the Pearson correlation coefficient calculated?

Answer 7

<correlate> or <pwcorr>
</pwcorr></correlate>

Question 8

What is the interpretation of a Pearson correlation coefficient of -0.43?

Answer 8

A moderate negative association

Question 9

What is the formula to calculate the CI for a slope (β1)?

Answer 9

β1 ± 1.96 x SE

Question 10

Why should the regression equation not be used outside the range of observed data?

Answer 10

Because the relationship outside the range may not be linear

Question 11

What does the R2 value represent in linear regression?

Answer 11

The proportion of variance in the dependent variable explained by the independent variable

Question 12

How do you compute a scatterplot?

Answer 12

<twoway(scatter varnames)>

Question 13

How do you compute a line of best fit?

Answer 13

<twoway(lfit varnames>

Question 14

How do you combine a scatterplot and line of best fit into one graph?

Answer 14

<twoway(scatter varnames) (lfit varnames)>

Question 15

Explain what 𝛽0 means in the context of the association between age and executive function
𝛽0 = 29.33
𝛽1 = -0.185

Answer 15

As the constant essentially sets the baseline when age = 0, executive function is 29.33. In other words, 𝛽0 (_const) describes the mean executive function at intercept between x and y, i.e., at the age of 0

Question 16

Is it meaningful to talk about individuals aged 0 in the context of the association between age and executive function?

Answer 16

No. To rectify this, we centre the exposure round the mean:
1. Find the mean of age and create a new variable (e.g., age_cent) where we subtract the mean from each value of x. Use the following commands:
<gen age_cent = age - r(mean)>
2. Draw a scatterplot and line of best fit using age_cent
3. The scale on the x axis should be different with data centred around the mean
4. 0 on the x axis now represents the mean age

Question 17

How do you do simple calculations in Stata?

Answer 17

<display> or <di> for short
</di></display>

Question 18

In the context of the association between age and executive function, after centring the exposure round the mean, how would you interpret the new _cons (𝛽0)>?
Mean age = 65
_cons = 17.26

Answer 18

When x = 65, executive function is 17.26

Question 19

How would you interpret a 95% CI of 0.2018 and 0.1691 and p-value of < 0.001
Context: Association between age and executive function
𝛽1: 0.18

Answer 19

We are 95% confident that the true decrease in executive function per additional year of age is between 0.1691 and 0.2018 points. The interval is narrow, suggesting the estimate of the slope is precise.
The p-value for 𝛽1 is < 0.001, thus there is strong evidence against H0. This suggests executive function declines with 0.18 per year increase in age.

Question 20

Outline the steps to computing a simple linear regression fitting sex against executive function:

Answer 20

1. Check how sex is coded (male = 1, female = 2)
2. Sex needs to be recoded so male = 0 and female = 1

<gen>
<recode sex01 (1=0) (2=1)>
<label define sex 0 "male" 1"female">
<label>
3. Visualise the data using this command:
<twoway(scatter var_names) (lfit var_names), xlabel (0 (1) 1)>
4. Fit a regression model using -
<regression>
</regression></label></gen>

Question 21

How do you compute a Pearson correlation?

Answer 21

<correlate>
or
<pwcorr>
</pwcorr></correlate>

Question 22

How do you compute a Spearman rank correlation?

Answer 22

<spearman>
</spearman>

Question 23

How do you formally interpret the results of a linear regression and correlation?

Answer 23

The results of a linear regression showed that (predictor) was a significant predictor of (outcome) (𝛽 = 𝛽1, 95% CI [a, b], p < 0.05). For each unit increase/decrease in (predictor), (outcome) increased/decreased by an estimated (𝛽1) points. The intercept (𝛽1) was (𝛽1) (95% CI [a, b], p < 0.05), representing the predicted (outcome) when (predictor) is equal to the mean (x). A Pearson correlation analysis indicated that the strength of the association was (strong/moderate/weak) (r)