1. Intervention Study: Kenya PRIMR Case Regression Analysis
March 2017
Susan Edwards, RTI International
Sarrynna Sou, RTI International

2. Overview Linear Regression Replicating T-Test Analysis
Replicating DiD Analysis
Controlling for Other Variables
Interpreting Estimates
Logistic Regression
STATA Code: Similar to Linear Regression
Interpreting Odds Ratios

3. Linear Regression Analysis
When would you want to use linear regression?
Used to Model Continuous Data
Examples:
Super Summary Variables: CLPM, ORF, etc.
Estimates Averages
General Form 𝑌=∝+ 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 …+ 𝜖 𝑖
Model Assumptions
Linearity
Observed Data are Fixed Constants
Errors are IID and N(0, σ)

4. Linear Regression Analysis - Example
Recall: 𝑌= ∝+ 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 …+ 𝜖 𝑖
orf = I(female) – 3.7 (age)
Interpretations:
This model suggests that girls read on average 2.7 words per minute more than boys when controlling for student age.
Assuming age is a continuous variable, for every year of age students have an average decrease of 3.7 words per minute when controlling for gender.
A 10 year old male student will read on average 75 – 3.7*10 = 38 words per minute.

5. Linear Regression Analysis – Reference Cell Coding
Recall: orf = I(female) – 3.7 (age)
Interpretations:
This model suggests that girls read on average 2.7 words per minute more than boys when controlling for student age.
Reference Cell Coding:
One level of a categorical variable is determined to be the reference.
All other estimates are presented in comparison to the reference.
Example:
Female 2 levels 0 = Male 1 = Female
2.7 I(female) = the # of wpm difference between males and females

6. Linear Regression Analysis – Reference Cell Coding
Example with More than 2 Levels:
Age Category 3 levels
0 = Younger than Grade Level = Below 7
1 = At Grade Level = 7 or 8
2 = Older than Grade Level = Above 8
Model for ORF:
ORF = I(At Grade Level) I(Older than Grade Level)
Questions:
What is the average fluency for students in public schools?
What is the average fluency for students in private schools?
Do students in public schools preform better on average than students in religious schools?
Do students in private schools preform better on average than students in religious schools?

7. Categorical vs. Continuous Independent Variables
Why do we care? STATA cares.
Categorical
Definition:
A variable that can be divided into distinct categories.
Examples:
gender
age category
STATA code:
Start variables with “i.” followed by variable name
i.<variable name>
Reference Cell Coding
Continuous
Definition:
A variable that theoretically could go on forever
Examples:
orf
age
Reading comprehension score?
Generally ranges from 0 to 5.
STATA code:
List variable name in equation line.

8. Linear Regression Analysis – STATA Example
Recall: 𝑌= ∝+ 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 …+ 𝜖 𝑖
orf = I(female) – 3.7 (age)
STATA Code:
svy: reg eq_orf i.female age

9. Linear Regression Analysis – STATA Activity
Recall: STATA code to fit a model for gender and age.
svy: reg eq_orf i.female age
Fit a linear model for English fluency (eq_orf) that accounts for the following school factors (nonformal; enrolment)
svy: reg eq_orf i.nonformal enrolment
Why does nonformal have an “i.” in front of the variable name?
What type of variable is enrolment in this model?
How would we change enrolment to be a categorical variable?
Would the model work if we typed the following?
svy: reg enrolment i.nonformal eq_orf

10. T-Test Results with Linear Regression in STATA
Recall: T-Tests compare the means of two groups.
Example: ttest eq_orf, by (treat_phase)
Is there a different between baseline and endline scores?
How can we use Linear Regression to duplicate these results?
October 2012
October 2013
Mean (N)
48 wpm (913)
53 wpm (922)
Difference (S.E.)
4.4 wpm (1.7)
T-Stat (DOF)
2.59 (1833)
H0: = ; Ha: !=
P-Value = ; Reject H0
Followed by hands on activity with Oral Reading Fluency.

11. T-Test Results with Linear Regression in STATA
Recall: ttest eq_orf, by (treat_phase)
How can we use Linear Regression to duplicate these results?
How many variables are in used in the ttest command?
eq_orf
treat_phase
Use a linear regression model that only contains the two variables of interest.
What would the STATA code for the model look like?
reg eq_orf i.treat_phase
Followed by hands on activity with Oral Reading Fluency.

12. T-Test Results with Linear Regression in STATA
Recall: ttest eq_orf, by (treat_phase) reg eq_orf i.treat_phase
October 2012
October 2013
Mean (N)
48 wpm (913)
53 wpm (922)
Difference (S.E.)
4.4 wpm (1.7)
T-Stat (DOF)
2.59 (1833)
H0: = ; Ha: !=
P-Value = ; Reject H0
Followed by hands on activity with Oral Reading Fluency.

13. T-Test Results with Linear Regression in STATA
Recall: ttest eq_orf, by (treat_phase) reg eq_orf i.treat_phase
October 2012
October 2013
Mean (N)
48 wpm (913)
53 wpm (922)
Difference (S.E.)
4.4 wpm (1.7)
T-Stat (DOF)
2.59 (1833)
H0: = ; Ha: !=
P-Value = ; Reject H0
These results are unweighted.
Wait!
How can we make our results reflect the population not the sample?
svy: reg eq_orf i.treat_phase
Followed by hands on activity with Oral Reading Fluency.

14. Linear Regression in STATA – Controlling for Other Variables
Want to Know:
Effect of certain variables when other variables we know to be influential are controlled.
Recall: orf = I(female) – 3.7 (age)
In this model, we may already know that older students are less fluent readers because they are repeating the grade or have taken a long break between school years. But we want to know if gender influences fluency once age is controlled.
When do we use models with multiple variables?
Determine Demographic and SSME Impact
What variables must be in these models?
Variables that we know strongly influence the outcome.
Sample design variables
Treatment; Gender; Time
Followed by hands on activity with Oral Reading Fluency.

15. Linear Regression in STATA – Controlling for Other Variables - Example
Fit a model for English fluency that accounts for treatment, time, gender, and formal/nonformal school type.
Question of Interest:
Once design variables are controlled for, is there a difference between students in formal and nonformal schools?
STATA Code:
svy: reg eq_orf i.treatment i.treat_phase i.treatment#i.treat_phase i.female i.nonformal
Interpretation:
Students in nonformal schools read on average 29 wpm more than students in formal schools when study design is controlled.
Followed by hands on activity with Oral Reading Fluency.

16. Linear Regression in STATA –
Linear Regression in STATA – Controlling for Other Variables - Activity
Activity:
Determine if any of the other SSME variables make a difference on student English reading fluency (eq_orf).
Followed by hands on activity with Oral Reading Fluency.

17. Linear vs Logistic Regression
When would you want to use logistic regression?
Used to Model Binomial Categorical Data
Examples:
Zero Scores
0 = Score above Zero on Task
1 = Score equal Zero on Task
Reading Comprehension of 80% or Better
0 = Reading Comprehension Score < 80%
1 = Reading Comprehension Score >= 80%
Estimates Probabilities and Odds Ratios

18. Linear vs Logistic Regression
When would you want to use logistic regression?
Used to Model Binomial Categorical Data
Estimates Probabilities and Odds Ratios
General Form 𝑙𝑜𝑔𝑖𝑡 𝑝 = log 𝑝 𝑖 1− 𝑝 𝑖 = 𝛽 0 + 𝛽 1 𝑥 1 ,
𝑤ℎ𝑒𝑟𝑒 𝑝=% 𝑜𝑓 𝑠𝑢𝑐𝑐𝑒𝑠𝑠= exp 𝑚𝑜𝑑𝑒𝑙 1+exp(𝑚𝑜𝑑𝑒𝑙)
Model Assumptions
Data are from a stratified SRS
Independence of responses between respondents.
Sample size is large; 80% of predicted counts at or about 5; all expected counts are larger than 2
Model is specified correctly

19. Linear vs Logistic Regression – Covariates & Odds Ratios
Covariates Connected to Odds Ratios
Example:
Reading Comprehension 80%+ =
I(Has English Book)
Odds Ratio:
English Book vs. No English Book = exp(0.76) = 2.14
Interpretation:
On average students with English books will be 2 times more likely than students without English books to comprehend at least 80% of a connected text.

20. Linear vs Logistic Regression – Covariates & Probabilities
Covariates Connected to Probabilities
Example:
Reading Comprehension 80%+ =
I(Has English Book)
Probability:
Pr(80%+ | English book) = exp − exp(−3+0.76) = 0.098
Interpretation:
On average a student with an English book is 9.8% likely to comprehend at least 80% of a passage.

21. Logistic Regression Analysis – STATA Example
Recall: Reading Comprehension 80%+ =
I(Has English Book)
NOTE: code is very similar to linear regression
STATA Code:
svy: logistic eq_read_comp_score_pcnt80 i.e_book
svy: logistic eq_read_comp_score_pcnt80 i.e_book, coef
Why does e_book have an “i.” in front of the variable name?
Why doesn’t eq_read_comp_score_pcnt80 have an “i.”?
What is the difference between the two lines of code?

22. More Information
Susan Edwards Research Statistician
Sarrynna Sou Statistician