1. Unit 6: The basics of multiple regression Class 14… Class 15…
© Andrew Ho, Harvard Graduate School of Education

2. Where is Unit 6 in our 11-Unit Sequence?
The basics of
multiple regression
Unit 7:
Statistical control in depth:
Correlation and collinearity
Unit 10:
Interaction and
quadratic effects
Unit 8:
Categorical predictors I: Dichotomies
Unit 9:
Categorical predictors II: Polychotomies
Unit 11:
Regression in practice.
Common Extensions.
Unit 1:
Introduction to
simple linear regression
Unit 2:
Correlation
and causality
Unit 3:
Inference for the
regression model
Building a
solid
foundation
Unit 4:
Regression assumptions:
Evaluating their tenability
Unit 5:
Transformations
to achieve linearity
Mastering
the
subtleties
Adding additional predictors
Generalizing to other types of predictors and effects
Pulling
it all
together
© Andrew Ho, Harvard Graduate School of Education

3. In this unit, we’re going to cover…
Various representations of the multiple regression model:
An algebraic representation
A three dimensional graphic representation
A two dimensional graphic representation
Multiple regression—how it works and helps improve predictions
Estimating the parameters of the multiple regression model
Holding predictors constant—what does this really mean?
Plotting the fitted multiple regression model:
Deciding how to construct the plot
Choosing prototypical values
Learning how to actually construct the plot (and interpret it correctly!)
𝑅2 and the Analysis of Variance (ANOVA) in multiple regression
Inference in multiple regression
The omnibus 𝐹-test in multiple regression
Individual 𝑡-tests
How might we summarize MR results in both tables and figures?
© Andrew Ho, Harvard Graduate School of Education
SEGUE: To do this, we need to learn about hypothesis testing—which is the focus of this unit. But to set our work in context, let’s begin with an example

4. US News and World Report education school rankings from 2006
How do student characteristics like GRE scores and size of the doctoral class predict peer ratings of Ed Schools? Do schools gain in reputation for graduating large numbers of high-achieving students?
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5. As always, our starting point
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6. Scatterplot matrix: graph matrix
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7. Use graph combine to be more succinct.
Given the variable (enrollment) and the shape of these plots, a log transformation is worth exploring.
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8. Logarithmic transformation of the docgrad variable
Comparison of regression models predicting US News peer ratings from the size and log(size) of the doctoral cohort
n=87
Model A
on docgrad
Model B
on log(docgrad)
Estimated Slope
Estimated S.E.
𝑡 statistic
.687*
(.128)
5.36
27.274*
(5.649)
4.83 𝑅2
25.27%
21.52%
* p<.001
A marginal case. Residuals suggest the transformation results in a modest improvement. Keep the original docgrad variable to simplify interpretation.
© Andrew Ho, Harvard Graduate School of Education

9. From simple to multiple regression
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 = 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 = 𝛽 0 + 𝛽 1 𝑔𝑟𝑒+ 𝛽 2 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
More generally, let X1, X2, … Xk
represent k predictors
How does multiple regression help us?
Simultaneous consideration of many contributing factors
We explain more of the variation in 𝑌
Equivalently, more accurate predictions (smaller residuals)
Provides a separate understanding of each predictor, accounting for the effects of other predictors in the model (an attempt at holding the values of the other predictors constant)
Allows us to build models that can help support theories and hypotheses about causal and associative mechanisms.
© Andrew Ho, Harvard Graduate School of Education

10. The stata syntax: regress peerrate gre docgrad (𝑌, then 𝑋s)
Population Model:
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒= 𝛽 0 + 𝛽 1 𝑔𝑟𝑒+ 𝛽 2 𝑑𝑜𝑐𝑔𝑟𝑎𝑑+𝜖
Sample Prediction Equation:
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒+.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑
A 10-point increment in average GREs predicts a 6.14-point increment in peer ratings, assuming docgrad can be held constant. (Simple linear regression coefficient: 6.91).
A 10-student increment in the size of the graduating doctoral cohort predicts a 5.4-point increment in peer ratings, assuming gre can be held constant. (Simple linear regression coefficient: 6.87)
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11. © Andrew Ho, Harvard Graduate School of Education
From Unit 1: Three “best-fit” regression lines. The line you want, and why.
The OLS criterion minimizes the sum of vertical squared residuals.
Other definitions of “best fit” are possible:
Vertical Squared Residuals (OLS)
Horizontal Squared Residuals (X on Y)
Orthogonal Residuals
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12. Minimize vertical squared residuals to the best fit plane
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒+.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑
© Andrew Ho, Harvard Graduate School of Education

13. Regression decomposition from Unit 1
Week 1:
Regression decomposition from Unit 1
Plane to mean-plane +
Point to mean-plane =
Point to plane
∑ 𝑌 𝑖 − 𝑌 = ∑ 𝑌 𝑖 − 𝑌 ∑ 𝑌 𝑖 − 𝑌 𝑖 2
Sum of Squares Total (𝑆𝑆𝑇) = Sum of Squares Model (𝑆𝑆𝑀) + Sum of Squares Error (𝑆𝑆𝐸)
∑ 𝑌 𝑖 − 𝑌 2 ∑ 𝑌 𝑖 − 𝑌 2 = 𝑆𝑆𝑀 𝑆𝑆𝑇 = 𝑅 2
The proportion of total variation that is accounted for by the model.
What is SSE SST ? 𝑆𝑆𝐸 𝑆𝑆𝑇 = 𝑆𝑆𝑇−𝑆𝑆𝑀 𝑆𝑆𝑇 =1− 𝑆𝑆𝑀 𝑆𝑆𝑇 =1− 𝑅 2 …

14. Analysis of Variance regression decomposition in Stata
Week 1:
Analysis of Variance regression decomposition in Stata
= 𝑆𝑆𝑀 𝑆𝑆𝑇 =1− 𝑆𝑆𝐸 𝑆𝑆𝑇 = =58.22%
Analysis of Variance regression decomposition
∑ 𝑌 𝑖 − 𝑌 2 =∑ 𝑌 𝑖 − 𝑌 2 +∑ 𝑌 𝑖 − 𝑌 𝑖 2
Interpreting R2
58.22 percent of the variation in the peer ratings is “attributable to” or “accounted for by” or “explained by” or “associated with” or “predicted by” the size of the doctoral cohort and the GRE scores of the admits.
What about the remaining 41.78%?
Research, funding, famous people, location, history,
measurement error,
random error, individual variation, alien abductions…
Error is what we haven’t modeled yet.
175172 ≈
73190
The Ubiquitous 𝑹𝟐
The variance of 𝑌 that is accounted for by 𝑋…
The single most widespread and easily interpretable summary statistic derivable from a single regression analysis. Essential to describing the overall predictive function of the model.
© Andrew Ho, Harvard Graduate School of Education

15. Multiple regression supports inferences about “statistical control”
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒+.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑
A 10-point increment in average GREs predicts a 6.14-point increment in peer ratings, assuming 𝑑𝑜𝑐𝑔𝑟𝑎𝑑 can be held constant. (Simple linear regression coefficient: 6.91).
A 10-student increment in the size of the graduating doctoral cohort predicts a 5.4-point increment in peer ratings, assuming 𝑔𝑟𝑒 can be held constant. (Simple linear regression coefficient: 6.87)
Even accounting for one variable, the other variable still has predictive utility, and vice versa.
Let’s see what the model implications are for “holding docgrad constant” or “account/adjust for docgrad”… for a typical small school, a midsized school, and a large school.
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16. Distribution of the size of the doctoral cohort in 2004
Let’s pick a small school with a doctoral cohort of 20, a midsized school with a doctoral cohort of 45, and a large school with a doctoral cohort of 80.
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17. © Andrew Ho, Harvard Graduate School of Education
Conditional Regression Lines: Visualizing the multiple regression model
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒+.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑
Small school, 𝑑𝑜𝑐𝑔𝑟𝑎𝑑 = 20
Midsized school, 𝑑𝑜𝑐𝑔𝑟𝑎𝑑 = 45
Large school, 𝑑𝑜𝑐𝑔𝑟𝑎𝑑 = 80
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒 =− 𝑔𝑟𝑒
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒 = 𝑔𝑟𝑒
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒 = 𝑔𝑟𝑒
Large school
Midsized school
Small school
© Andrew Ho, Harvard Graduate School of Education

18. From simple to multiple regression
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 = 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 = 𝛽 0 + 𝛽 1 𝑔𝑟𝑒+ 𝛽 2 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
More generally, let X1, X2, … Xk
represent k predictors
How does multiple regression help us?
Simultaneous consideration of many contributing factors
We explain more of the variation in 𝑌
Equivalently, more accurate predictions (smaller residuals)
Provides a separate understanding of each predictor, accounting for the effects of other predictors in the model (an attempt at holding the values of the other predictors constant)
Allows us to build models that can help support theories and hypotheses about causal and associative mechanisms.
© Andrew Ho, Harvard Graduate School of Education

19. © Andrew Ho, Harvard Graduate School of Education
Our Data: Peer Rating
If each of you were a school of education…
With peer ratings, cohort sizes, and average GRE scores determined by your location in the 3D space of Larsen G08…
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20. © Andrew Ho, Harvard Graduate School of Education
Our Data: Cohort Size
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21. © Andrew Ho, Harvard Graduate School of Education
Our Data: Average GREs
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22. Our Data: Peer Rating on Cohort Size
This is clearly a weak, near zero relationship, far weaker than the empirical relationship between peer rating and cohort size.
How can you visualize this in this classroom?
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23. Our Data: Peer Rating on Average GRE
This is a weak-to-moderate relationship, slightly stronger than the empirical relationship between peer rating and average GRE scores.
How can you visualize this in this classroom?
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24. Multiple Regression: Finding the best-fit (hyper)plane
Vertical Squared Residuals (OLS)
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25. Multiple Regression on Our Data in Stata
𝑜𝑢𝑟𝑟𝑎𝑡𝑒 = 𝑜𝑢𝑟𝑠𝑖𝑧𝑒+.477𝑜𝑢𝑟𝑔𝑟𝑒
A 100-point increment in average GREs predicts a 47.7-point increment in peer ratings, assuming 𝑑𝑜𝑐𝑔𝑟𝑎𝑑 can be held constant.
A 100-student increment in the size of the graduating doctoral cohort predicts a 10.8-point increment in peer ratings, assuming 𝑔𝑟𝑒 can be held constant.
48.07% of the variation in ratings is accounted for by the two predictor variables.
On your own and in discussion and consultation with your neighbors:
Write down your name, and then estimate your PeerRate, DocGrad, and GRE values from your location in this room.
Use the prediction equation above and calculate your 𝑌 (your predicted peer rating, your fitted value).
Write down your residual. How can you interpret this residual?
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26. Constructing a conditional regression plot
- Select 2-5 prototypical values
- Substantively interesting values
- A range of percentiles (10 or 25, 50, 75 or 90)
- The sample mean ± .5 or 1 standard deviation
- Easily communicated values, whole numbers or fractions.
- When the variable is already ordinal (ordered categories, e.g., coded 0, 1, and 2 for small, medium, and large), so much the better
Large (80)
Midsized (45)
Small (20)
We can plot peerrate on gre with different docgrad lines, or we can plot peerrate on docgrad with different gre lines.
Generally, we place the primary predictor of interest on the 𝑋 axis (let’s say gre) and the “control” predictor or “covariate” on the legend.
With our data, we can pick prototypical oursize values of 20, 60, and 100.
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27. Conditional regression lines for our data
Large schools (oursize = 100)*
Small schools (oursize = 20)*
* Differences between conditional regression
lines are not statistically significant.
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28. Conditional regression lines for our data
High GRE Schools (ourgre = 700)*
Mid-Range GRE Schools (ourgre = 600)*
Low GRE Schools (ourgre = 500)*
* Slopes of conditional regression lines are not statistically significant (cannot be distinguished from 0).
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29. Interpreting the conditional regression plot for empirical data
Large (80)
Midsized (45)
Small (20)
We imagine that the plot extends into the slide along a third axis, docgrad, where larger schools are deeper and smaller schools are closer (or vice versa).
The regression model fits the best fit *plane* through this scatterplot in three dimensional space, minimizing the vertical squared residuals between each point and the plane.
Instead of a single regression line of peerrate on gre, we have a regression line for every level of docgrad, extending along the plane.
Note that the lines are parallel, and they must be given our regression model: 𝑌= 𝛽 0 + 𝛽 1 𝑋 1 + 𝛽 2 𝑋 2 +…+ 𝛽 𝑘 𝑋 𝑘 +𝜖 We loosen this assumption in Unit 10.
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30. Interpreting a conditional regression plot
Adding the other predictor improves prediction, reduces residuals, and reduces residual variance, but residual variance obviously remains.
The magnitude of the axis predictor can be seen as the slope. The magnitude of the legend predictor can be seen as the spacing between the lines.
Large (80)
Midsized (45)
Small (20)
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31. Illustrative contrasts in regression lines
Prediction of peer rating by average GRE scores for large schools (docgrad = 80)
Prediction of peer rating by average GRE scores without accounting for school size. The unconditional regression line.
This is not the prediction for an average-sized school.
This is the prediction if we knew nothing about size!
Prediction of peer rating by average GRE scores for the average-size schools (docgrad = 45.68). The conditional regression line.
Prediction of peer rating by average GRE scores for small schools (docgrad = 20)
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32. Another perspective on the implications of the same model…
Let’s pick a low-GRE school of 500, a midrange GRE school of 550, and a high GRE school of 600.
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33. Another perspective on conditional regression lines
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− 𝑔𝑟𝑒+.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑
Low-scoring school, 𝑔𝑟𝑒 = 500
Midrange school, 𝑔𝑟𝑒 = 550
High-scoring school, 𝑔𝑟𝑒 = 600
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− (500) +.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑= 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− (550) +.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑= 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
𝑝𝑒𝑒𝑟𝑟𝑎𝑡𝑒 =− (600) +.540𝑑𝑜𝑐𝑔𝑟𝑎𝑑= 𝑑𝑜𝑐𝑔𝑟𝑎𝑑
A 10-person increment in the size of the doctoral cohort predicts a 5.4-point increment in the predicted mean peer rating by deans, assuming average GREs can be held constant.
An 50-point increment in average GRE scores predicts a 30.7-point increment in the predicted mean peer rating by deans, assuming the size of the doctoral cohort can be held constant.
High-scoring school (600)
Midrange school (550)
Low-scoring school (500)
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34. © Andrew Ho, Harvard Graduate School of Education
Week 1:
ANalysis Of VAriance (ANOVA) regression decomposition for multiple regression
MS stands for Mean Square, an average sum of squares, like a variance.
In fact, MS Total *is* your 𝑌 variance, the unconditional variance of the peerrate variable, the square of the standard deviation: 𝑌 𝑖 − 𝑌 2 𝑛−1 =
MS Model, the model mean square, is a measure of the variation accounted for by the model (not easily interpretable on the 𝑌 scale). The bigger the better.
MS Residual, your residual variance, is interpretable as error variance (𝑅𝑀𝑆𝐸2)
Analysis of Variance
Total
Residual
(Error)
Model (Regression)
Sum of Squares
Source
Everything to this point is the same as simple linear regression, except that we are paying more attention to the mean squares.
MST=SST/dfSST
n – 1
MSE=SSE/dfSSE
n – k – 1
MSR=SSR/dfSSR
k ~ #predictors
Mean Square df
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35. The 𝐹 distribution, a sampling distribution for variance ratios
Analysis of Variance
Total
Residual
(Error)
Model (Regression)
Sum of Squares
Source
MST=SST/dfSST
n – 1
MSE=SSE/dfSSE
n – k – 1
MSR=SSR/dfSSR
k ~ #predictors
Mean Square df
F distribution:
Sampling distributions:
If we sample two variances (MSR and MSE) from two populations with equal variances, and if we take repeated ratios (MSR/MSE), sometimes the numerator (MSR) will be larger, and sometimes the denominator (MSE) will be larger.
The distribution will be loosely centered on 1 and will always be positive.
If one variance is craaaaazy bigger than another, we might conclude that our null hypothesis of equal population variances is incorrect and accept the alternative: unequal population variances.
This is the sample size from the first distribution (-1)
This is the sample size from the second distribution (-1)
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36. The omnibus 𝐹 test for multiple regression
Analysis of Variance
Total
Residual
(Error)
Model (Regression)
Sum of Squares
Source
MST=SST/dfSST
n – 1
MSE=SSE/dfSSE
n – k – 1
MSR=SSR/dfSSR
k ~ #predictors
Mean Square df
MSR and MSE are scaled (by dividing by 𝑑𝑓) such that, under the null hypothesis, they act like sample variances from populations with equal variance.
This particular 𝐹 statistic represents variance accounted for by the regression model over error variance (unaccounted for).
In other words, good variance over bad variance. We want F to be as large as possible.
Under the null hypothesis, there is no predictive value to *any* of your predictors in the population:
If the model variance is sufficiently greater than the error variance, we can reject
We accept One or more population slopes are nonzero.
This is 𝑘, the number of predictors
This is 𝑛−𝑘−1, a quantity that increases with sample size
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37. © Andrew Ho, Harvard Graduate School of Education
The omnibus 𝐹 test
Null hypothesis: The predictors account for no Y variance in the population. The predictors have no predictive utility in the population.
Test statistic: 𝐹 = 𝑀𝑆𝑅/𝑀𝑆𝐸. Ratio of regression model variance to error variance. Variance accounted for over variance unaccounted for.
Decision rule: If 𝐹 > 𝐹𝑐𝑟𝑖𝑡, reject 𝐻0 in favor of 𝐻𝑎: one or more population slopes are nonzero; the model has some predictive utility, some 𝛽 𝑗 ≠0
Here 𝑗 indexes one of 𝑘 predictor slopes.
Critical values of 𝑭 (α=.05)
1000
120 20 10 5 4 3 2 1
𝑑𝑓 for numerator (MSR)
inf
100 50 25
𝑑𝑓 for denominator (MSE)
Stata returns 𝐹 critical values as:
display invFtail(k, n-k-1, .05)
1.00
1.30
1.56
1.72
1.22
1.38
1.64
1.77
1.57
1.68
1.78
2.01
1.83
1.93
2.03
2.24
2.21
2.31
2.40
2.60
2.37
2.46
2.56
2.76
2.70
2.79
2.99
3.00
3.09
3.18
3.39
3.84
3.94
4.03
4.24
k ~ number of predictors
n-k-1
The probability of sampling an 𝐹 ratio of or larger under the null hypothesis of no predictive utility is … very, very low, so we reject the null hypothesis and conclude that one or more predictors account for some variance in the population.
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38. The omnibus F test vs. slope tests
Omnibus F test Across all of my predictors, is regression helping at all in the population?
t-tests for slopes
In the population, does this variable help prediction above and beyond other included variables?
In the population, regression does not help at all
In the population, this set of predictors has some predictive utility
In the population, this predictor has no effect when accounting for other predictors in the model.
In the population, this predictor has an effect when accounting for other predictors in the model.
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39. Three equivalent tests in simple linear regression
Let’s revisit a simple linear regression model analogous to one you’re considering in Assignment #3: Math Achievement (TIMSS) on the log of income (per capita GDP) for countries.
Fun fact 2: When k = 1, R-sq = 𝑟2
Test 1: The omnibus 𝐹 test Evaluates whether the any of the populations slopes are nonzero. With only one slope…
Test 2: The 𝑡-test for slope
Evaluates whether a particular population slope is nonzero…
Test 3: The 𝑟-test for correlation (pwcorr command)
Evaluates whether a population correlation is nonzero.
Fun fact 1: When there is only one predictor (that is, the degrees of freedom in the numerator is 1), then 𝐹= 𝑡 2 , 4.97≈ and the 𝑝-values from the two tests will be identical (.0315).
With a single predictor, the significance of the omnibus prediction, the significance of a single slope, and the significance of an association are identical.
With k > 1 predictor, this equivalence does not hold.
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40. Summarizing regression output from multiple models: estout
Find and install the estout package with the following command: findit estout and scroll past the search results to the third listed package under the Web Resources heading. Click the link, then click the helpful link to the right, labeled, “Click here to install.” As always, you will begin by conducting a thorough review of the univariate and bivariate distributions and statistics, exploring various models and diagnostics, and considering remedial options. At a certain point, you may consider saving promising, helpful, illustrative or otherwise benchmark-setting models for your review. This can be done with the estout package Documentation:
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41. Important esttab options
Add a descriptive title and short but descriptive coeflabels. Descriptive model titles (mtitles) should be added for reference and numbers suppressed (nonumbers). With many models, you may need the compress option, although this will limit the length of your coeflabels and mtitles. APA guidelines require that figures and tables be self-contained, so a table note (addnote) can be useful. Finally, note our desired statistics, R-sq, adj-R-sq, F, and our degrees of freedom for the model and for the residual.
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42. Other esttab features to note
If you store too many models with eststo, clear its memory with eststo clear. Then you’ll have use eststo quietly regress … to store again from zero.
Never forget the importance of help, e.g., help esttab
When presenting tables, it is generally more common to include standard errors, not t statistics, in parentheses. To do this, use the se option.
Likewise, for parsimony, the F statistic and degrees of freedom are not commonly reported, although they can be helpful for model selection.
The R-sq is a necessity. The adjusted R-sq helps in model selection (you’ll soon see why), but it is less commonly reported unless model selection is the focus.
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43. Reporting results for the multiple regression model
Reporting statistical significance:
The two-predictor model with average GRE score and the size of the doctoral cohort accounts for 58.2% of the variation in peer ratings.
The omnibus null hypothesis of no predictive utility can be rejected F(2,84)=58.52, p<.001.
(The omnibus p-value is not shown in this table and must be gathered from the original regression output as Prob>F.)
The coefficient for the average GRE score is statistically significant, t(84)=8.14, p< The model suggests that an increment of 10 points in average GRE score is associated with an increment of 6.14 points on the peer rating scale, if the size of the doctoral cohort could be held constant.
The coefficient for the size of the doctoral cohort is statistically significant, t(84)=5.51, p< The model implies that an increment of 10 students in doctoral degree conferral predicts an increment of 5.40 points on the peer rating scale, accounting for the average GRE score of the students.
© Andrew Ho, Harvard Graduate School of Education

44. Beginning to look across the models
Each predictor’s estimated coefficient decreases in magnitude when the other predictor is included in the model.
However, the signs are unchanged, the decline does not appear to be substantively significant, and the statistical significance of the two predictors remains unchanged.
Our inferences about the prediction of peer ratings are reasonably robust across the model forms and compositions shown in this table.
Given that our interpretations are robust to model choice, and the two-predictor model is substantively defensible, we opt for the model that supports the best predictions.
The average GRE score alone accounts for 43.1% of the variance in peer ratings, and the size of the graduating doctoral cohort alone accounts for 25.3% of the variance in peer ratings.
Together, these two variables account for 58.2% of the variance in peer ratings.
Note that 43.1% % = 68.4% of the variance, yet only 58.2% of the variance is accounted for by both variables. Why the difference?
The percentages will sum tidily if and only if the predictors have zero correlation with each other, a veritable impossibility with real data outside of a coding error.
© Andrew Ho, Harvard Graduate School of Education

45. Diagnostic plots for multiple regression
Our regression assumptions haven’t changed: independent, normally distributed residuals with equal variance centered on 0. Our plots remain the same, as well: residuals vs. fitted values (rvfplot), leverage vs. discrepancy (lvr2plot), and Cook’s distance, to name a few.
We speak now in terms of discrepancy from, leverage on, and influence upon the estimated regression plane.
© Andrew Ho, Harvard Graduate School of Education

46. © Andrew Ho, Harvard Graduate School of Education
Midterm Checkin
Assignment #3 will be passed back by over the next 24 hours.
Assignment #4 has been posted, partners mandatory.
Notify me by Friday if you do not have a partner. No exceptions.
Partnerships take work. Open communication. Constant upkeep.
Attendance. Come to class regularly.
Resources: Lectures, videos, sections, partnerships, study groups, my office hours.
Course Philosophies: Disciplined Perception. Layering. Language. Collaboration. Sensitivity. Statistical vs. Substantive. Exploratory Analysis.
© Andrew Ho, Harvard Graduate School of Education

47. What are the takeaways from this unit?
Multiple regression holds several advantages over single-predictor regression
Simultaneous consideration of multiple correlated predictors
Improved prediction of outcomes
Parsimony in model expression
Support of inferences like, “the effect of X while accounting for Z”
More control over models consistent with our theories and hypotheses
Construction and conceptualization of 2D and 3D representations of two-predictor models in terms of best-fit planes
ANOVA decomposition and R-sq for multiple regression
The omnibus F test vs. t-tests for slopes
The estout package and the tabular juxtaposition of models
Regression diagnostics in multiple regression
© Andrew Ho, Harvard Graduate School of Education