1. Applied Biostatistics: Lecture 2
Brian Healy, Ph.D

2. Goals At the end of this lecture, you will be able to
Perform a two sample comparison with a continuous outcome in R
Perform a linear regression analysis in R
Assess the assumptions of linear regression in R

3. Review In the previous class, we focused on: Reading data into R
Calculating summary statistics
Creating graphics
Performing a chi-squared test

4. Results for review from previous class
Let’s import the dataset from last week
To assess the treatment effect on death, we can use the following commands:
chisq.test(table(rct$group,rct$death), correct=F)
prop.test(table(rct$group,rct$death), correct=F)
What happens if we use this command:
prop.test(table(rct$death,rct$group), correct=F)

5. Example
For today’s class, we will focus on investigating whether there is a difference between male MS patients and female MS patients in terms of cognitive functioning
The dataset we will use is the Practice_data.csv dataset from the course website

6. Variables
We can see the variables in the dataset using the names command:
names(data_prac)
From this command, we can see that there seven variables in our dataset
male: indicator variable of whether the subject is male or female (=1 for male; =0 for female)
age: subject’s age
dep: subject’s depression score
mcs, pcs: subject’s mental and physical quality of life
fatig: subject’s fatigue score
sdmt: subject’s score on the symbol digit modalities test

7. Roles of variables Outcome variable: Explanatory variable:
Symbol Digit Modalities Test (SDMT) is a brief measure of cognitive functioning that measures speed of information processing
This is a continuous variable
Explanatory variable:
Male
This is a dichotomous variable
Other variables
Age, fatigue score
These are continuous variables that could influence the analysis

8. Descriptive statistics
For a dichotomous variable, the descriptive statistics of interest are the number of subjects and the proportion
table(data_prac$male)
prop.table(table(data_prac$male))
For a continuous variable, the descriptive statistics of interest are the mean and standard deviation
mean(data_prac$sdmt)
sd(data_prac$sdmt)

9. Graphics
For a continuous variable, it is also a good idea to plot the data
Some plots look only at the outcome
hist(data_prac$sdmt)
boxplot(data_prac$sdmt)
We can also plot the data along with predictor variables
boxplot(data_prac$sdmt~data_prac$male)
plot(x=data_prac$age,y=data_prac$sdmt, ylab="SDMT", xlab="Age (years)")

10. Study design
In the previous lecture, we investigated the effect of pravastatin on specific events based on a randomized clinical trial (RCT)
In an RCT, subjects are randomized to treatment group, which balances the groups on measured and unmeasured confounders
We will infer that any statistically significant difference between the groups is due solely to the treatment
Today, we are performing an analysis of an observational study
We can’t randomize subjects to gender so we could observe differences for reasons other than gender
A simple two group comparison might not be enough to analyze the data

11. Univariate analysis
The next step in this analysis will be to assess if there is an association between gender and SDMT score
Summary statistics in each group
Males
mean(data_prac[data_prac$male==1,]$sdmt)
sd(data_prac[data_prac$male==1,]$sdmt)
Females
mean(data_prac[data_prac$male==0,]$sdmt)
sd(data_prac[data_prac$male==0,]$sdmt)
Based on these statistics, there seems to be not much difference between the groups

12. Two sample t-test
Despite the limited difference between the groups, we would like to formally test if there is a difference
To test for a difference between groups with a continuous outcome, we use a two sample t-test
For the two sample t-test, we compare the mean SDMT in the males to the mean SDMT score in the females accounting for the variability in the data and the sample size
If there was no difference, what would we expect the difference in the means to equal?

13. Formula Basic form of test statistic:
Is the observed mean difference similar to the hypothesized mean difference relative to the variability in the data?
The denominator is calculated differently based on whether we assume equal variance in the two groups

14. Two sample t-test in R
To perform a two sample t-test in R, we use the following command:
t.test(data_prac$sdmt~data_prac$male)
The default in R is the unequal variance version of the two sample t-test
This test is valid even if the variance in the two groups are not equal
We can fit the equal variance t-test using this command:
t.test(data_prac$sdmt~data_prac$male,var.equal=T)
We will focus on the equal variance t-test for now

15. Steps for hypothesis testing
State null hypothesis
State type of data for explanatory and outcome variable
Determine appropriate statistical test
State summary statistics if possible
Calculate p-value (stat package)
Decide whether to reject or not reject the null hypothesis
Write conclusion

16. Hypothesis test H0: meanfemales =meanmales
Explanatory: dichotomous, outcome: continuous
Two sample t-test
Estimated mean (mêanfemales) =49.2,
Estimated mean (mêanmales)=47.0
Calculate p-value: p = 0.23
Fail to reject the null hypothesis because p-value is greater than 0.05
Conclusion: The difference between the groups is not statistically significant.

17. Confidence interval
In addition to the p-value from the hypothesis test, we might also like to estimate the difference in the means between the groups
The point estimate is the difference in the group means
Difference in group means=2.2
How should we interpret the positive sign?
The confidence interval provides a range of plausible values for the difference in the means
The t.test command provides this confidence interval as part of the output
How should we interpret this confidence interval?

18. What happened here? Here is another t-test command:
t.test(data_prac$sdmt, data_prac$male)
What is the difference between this command and the previous command?
Why was this command not appropriate?

19. Linear regression
An alternative approach to compare the groups is to use linear regression
Linear regression can be used to analyze a continuous outcome with dichotomous, continuous or categorical predictors
The linear regression model to assess the association between gender and SDMT is

20. Indicator variables
In the previous model, the variable male is equal to 0 for the females and equal to 1 for the males
For the females, the model is:
E(SDMTi|malei=0)=b0
For the males, the model is:
E(SDMTi|malei=1)=b0+ b1
How should we interpret each coefficient?
If there was no difference between the genders, what would b1 equal?

21. Fitting linear regression in R
There are two equivalent ways to fit a linear regression model with an indicator variable
Since the variable male is already equal to 0 or 1, we can enter it into the model directly
model<-lm(sdmt~male,data=data_prac)
This command stores the results of the linear regression under the name model
Alternatively, we can tell R that male is a dichotomous/categorical variable using the factor notation
model2<-lm(sdmt~factor(male),data=data_prac)

22. Linear regression results
To print the results from R including the estimated coefficients and p-values, we use the summary command
summary(model)

23. Hypothesis test H0: meanfemales =meanmales H0: b1 =0
Explanatory: dichotomous, outcome: continuous
Linear regression
Estimated b1 =-2.2
Calculate p-value: p = 0.23
Fail to reject the null hypothesis because p-value is greater than 0.05
Conclusion: The difference between the groups is not statistically significant.

24. Confidence interval
To get the confidence interval for the coefficients, we can use the confint command applied to model
This provides the confidence interval for both of the beta coefficients
Does the confidence interval for male include the null value?

25. Try on your own
In addition to assessing the association between gender and SDMT scores, we might be interested in the effect of age on SDMT scores
Fit a linear regression in R with SDMT score as the outcome and age as the predictor
What was the estimated regression equation?
Was age a statistically significant predictor?
How did SDMT score change with increasing age?

26. Additional variables
Although we did not see an association in the univariate analysis of gender, this might not be the end of the story since we did not have a randomized trial
We know that age and fatigue can have an important impact on SDMT scores
If there is a difference between the males and females in terms of age or fatigue, we would want to control for these in the analysis

27. Multiple regression
The most common approach for handling multiple predictors in the same model is using multiple regression
Model
Multiple regression includes extra predictors on the right side of the equation
All coefficients must be interpreted based on all of the features in the model

28. Interpretation of coefficients
For this model,
b0 is the mean SDMT score when male, age and fatigue are all equal to 0
b1 is the change in the mean SDMT comparing males to females, HOLDING age constant
b2 is the change in the mean SDMT for a one-unit increase in age, HOLDING male constant

29. Multiple linear regression in R
To fit this model in R, we can use the same command adding the additional variable
model3<-lm(sdmt~male+age,data=data_prac)
summary(model3)

30. Hypothesis test
H0: There is no difference between males and females controlling for age
H0: b1 =0
Explanatory: dichotomous, outcome: continuous
Linear regression
Estimated b1 =-2.3
Calculate p-value: p = 0.20
Fail to reject the null hypothesis because p-value is greater than 0.05
Conclusion: The difference between the groups controlling for age is not statistically significant.

31. Confidence interval
To get the confidence interval for the coefficients, we can use the confint command applied to model
This provides the confidence interval for both of the beta coefficients
Does the confidence interval for male include the null value?

32. Regression assumptions
Linear regression has four main assumptions
Independence
Linearity/correct model
Homoscedasticity of residuals
Normality of residuals
Each of the final three can be investigated by looking at regression diagnostic plots
For more information about regression diagnostics, please see

33. Regression diagnostics
R also easily allows us to create a set of regression diagnostic plots
par(mfrow=c(2,2))
plot(model3)

34. Plots
In addition to the residual plots, we might want to plot the relationship between age and SDMT score with different symbols for the males and females
plot(x=data_prac$age, y=data_prac$sdmt, type="n", xlab="Age (years)", ylab="SDMT")
points(x=data_prac[data_prac$male==0,]$age, y=data_prac[data_prac$male==0,]$sdmt, col="red")
points(x=data_prac[data_prac$male==1,]$age, y=data_prac[data_prac$male==1,]$sdmt, col="blue")

35. Add fitted lines
We can add the fitted lines to the graph using these command
data_prac$predval<-predict(model3)
data_prac<-data_prac[order(data_prac$age),]
lines(x=data_prac[data_prac$male==0,]$age, y=data_prac[data_prac$male==0,]$predval, col="red")
lines(x=data_prac[data_prac$male==1,]$age, y=data_prac[data_prac$male==1,]$predval, col="blue")
What can we say about the fitted lines?

36. Try on your own
Fit a linear regression in R with SDMT score as the outcome and male, age, and fatigue as the predictors
What was the estimated regression equation?
Was male a statistically significant predictor in this new model?

37. Interaction
In the previous model, we assumed that the lines in the two genders were parallel
This means that the change with age was the same in the two genders
We can relax this assumption by adding an interaction term

38. Interpretation of coefficients
Females:
Males:
b3 is the difference in the slope comparing males and females
If b3=0, what would that mean?

39. Multiple linear regression in R
To fit this model in R, we can use the same command adding the additional variable
model4<-lm(sdmt~male+age+male:age, data=data_prac)
summary(model4)

40. Hypothesis test
H0: There is no interaction between age and gender in terms of the effect on SDMT score
H0: b3 =0
Explanatory: dichotomous, outcome: continuous
Linear regression
Estimated b1 =-0.27
Calculate p-value: p = 0.097
Fail to reject the null hypothesis because p-value is greater than 0.05
Conclusion: The interaction is not statistically significant.

41. Caution
Care must be taken when fitting interaction terms because the meaning of the other coefficients has changed
b1 is now the difference between the males and females when age=0
b2 is now the effect of a one unit in age in the females
You must be careful to understand the meaning of your coefficients

42. Goals At the end of this lecture, you will be able to
Perform a two sample comparison with a continuous outcome in R
Perform a linear regression analysis in R
Assess the assumptions of linear regression in R