Multiple Regression

Objectives

𝑦 = 𝛽

0

+ 𝛽

1

𝑋

1

+ ⋯ + 𝛽

𝑝

𝑋

𝑝

•

Maximize the predictive power of the independent variables as represented in the variate.

•

Compare two or more sets of independent variables to ascertain the predictive power of each variate

Explanation

•

The most direct interpretation of the regression variate is a determination of the relative importance of each independent variable in the prediction of the dependent measure.

•

Assess the nature of the relationships between the independent variables and the dependent variable.

•

Provide insight into the relationships among independent variables.

X 3

Y’

X 1 X 2

Sample Problem (Leslie Salt Property):Finding Fair Price of a Land

Variable Name

PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE

Description

Sale price in $000 per acre San Mateo=0, Santa Clara=1 Size of the property in acres Average elevation in feet above sea level Distance (in feet) to nearest sewer connection Date of sale counting backward from current time (in months) Subject to flooding by tidal action =1; otherwise=0 Distance in miles from Leslie property

19.3

11.7

13.3

15.1

12.4

15.3

12.2

18.1

16.8

5.9

4 37.2

18.2

15.1

22.9

15.2

21.9

PRICE

4.5

COUNTY SIZE

1 138.4

ELEVATION SEWER

10 3000

DATE

-103

FLOOD

10.6

1.7

1 0 52 16.1

4 0 0 2640 -103 -98 5 5 3.3

5.7

0 0 1 1 1695.2

845 6.9

105.9

1 1 2 4 3500 1000 10000 0 -93 -92 -86 -68 6.2

19.4

3.2

4.7

6.9

8.1

11.6

1 1 1 1 1 1 1 56.6

51.4

22.1

22.1

27.7

18.6

69.9

4 20 0 0 3 5 8 0 1300 6000 6000 4500 5000 0 -64 -63 -62 -61 -60 -59 -59 1 1 1 1 1 1 0 1 1 0 0 0 0 0 0 0 0 145.7

77.2

26.2

102.3

49.5

12.2

320.6

9.9

15.3

55.2

116.2

15 23.4

132.8

12 67 30.8

10 9 8 6 11 8 0 5 2 0 2 5 5 2 5 2 2 0 0 0 0 0 0 4000 0 0 1320 900 0 4420 2640 3400 900 900 -59 -59 -59 -59 -59 -59 -54 -54 -53 -49 -45 -39 -39 -35 -16 -5 -4 4.2

4.5

4.7

4.9

4.6

5 16.5

5.2

5.5

11.9

5.5

7.2

5.5

10.2

5.5

5.5

5.5

0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0

DISTANCE

0.3

0 1 2.5

10.3

14 14 0 0 0 1.2

0 0 0 0.5

4.4

Histogram of leslie_salt[, 1]

0 10 20 leslie_salt[, 1] 30

Histogram of Y

40 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

Y

SEWER FLOOD SIZE COUNTY DISTANCE ELEVATION DATE PRICE

PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE PRICE COUNTY SIZE

100.00% -18.22% -23.97% -18.22% 100.00% -33.94% -23.97% -33.94% 100.00%

ELEVATION SEWER

35.18% -39.12% 47.52% -20.95%

DATE FLOOD

59.47% -32.31% -5.00% -36.98% -55.18% -74.22% 5.34% -34.95% 10.89%

DISTANCE

9.33% 55.69% 35.18% -39.12% 47.52% -20.95% -5.00% 59.47% -36.98% -34.95% -32.31% -55.18% 9.33% -74.22% 5.34% 10.89% 55.69% 100.00% -35.94% -37.31% -11.31% -36.25% -15.87% -5.65% -37.31% -36.25% -35.94% 100.00% -15.15% -11.31% -15.87% -5.65% -15.15% 100.00% 4.44% 1.54% 1.54% 100.00% 4.44% 42.33% 42.33% 100.00%

0.5

PRICE 0.0

0.4

0.8

COUNTY 2.0

3.5

0 5 15 -100 -60 -20 0 5 10 15 SIZE 0 1000 ELEVATION SEWER 0 4000 10000 DATE FLOOD 0.0

0.4

0.8

DISTANCE

PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE PRICE

100.00% -18.22% -23.97%

COUNTY

-18.22% 100.00% -33.94%

SIZE

-23.97% -33.94% 100.00%

ELEVATION

35.18% 47.52% -20.95%

SEWER

-39.12% -5.00% 5.34%

DATE

59.47% -36.98% -34.95%

FLOOD

-32.31% -55.18% 10.89%

DISTANCE

9.33% -74.22% 55.69% 35.18% -39.12% 59.47% -32.31% 9.33% 47.52% -5.00% -36.98% -55.18% -74.22% -20.95% 5.34% -34.95% 10.89% 55.69% 100.00% -35.94% -5.65% -37.31% -36.25% -35.94% 100.00% -15.15% -11.31% -15.87% -5.65% -15.15% 100.00% 1.54% 4.44% -37.31% -11.31% 1.54% 100.00% 42.33% -36.25% -15.87% 4.44% 42.33% 100.00% log 𝑃𝑅𝐼𝐶𝐸 = 𝑏 0 + 𝑏 1 𝐸𝐿𝐸𝑉𝐴𝑇𝐼𝑂𝑁 + 𝑏 2 𝑆𝐸𝑊𝐸𝑅 + 𝑏 3 𝐷𝐴𝑇𝐸 + 𝑏 4 𝐹𝐿𝑂𝑂𝐷 + 𝜀

summary(model) Call: lm(formula = leslie_salt[, 1] ~ leslie_salt[, 4] + leslie_salt[, 5] + leslie_salt[, 6]) Residuals: Min 1Q Median 3Q Max -9.6076 -3.2506 -0.0281 2.8770 20.2776 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 21.2787636 2.9203157 7.286 7.75e-08 *** leslie_salt[, 4] 0.5614588 0.2515472 2.232 0.034107 * leslie_salt[, 5] -0.0005871 0.0004460 -1.316 0.199129 leslie_salt[, 6] 0.1836824 0.0421712 4.356 0.000172 *** -- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.559 on 27 degrees of freedom Multiple R-squared: 0.5327, Adjusted R-squared: 0.4807 F-statistic: 10.26 on 3 and 27 DF, p-value: 0.000111

Assumptions

•

Linearity of the dependent variable in terms of independent variables.

• • 𝑦 = 𝛽 0 𝑦 = 𝛽 0 + 𝛽 1 𝑋 1 2 + 𝛽 1 𝑒 𝑋 1 + ⋯ + ⋯ 𝑦 • 𝑦 = 𝛽 0 + 𝛽 1 sin 𝑋 1 + ⋯ 𝑋 1

Linearity (cts.)

A higher order term of the dependent variable should be included.

In that case define a new variable by taking the square (for this case) of that independent variable and use squared values in the regression.

Use: Visual inspection

More troublesome is MODERATOR effect

•

If an independent-dependent variable relationship is effected by another independent variable this situation is termed a moderator effect .

•

The most common moderator effect in multiple regression is the bilinear moderator in which the slope of the relationship of one independent variable (X 1 ) changes across values of the moderator variable (X 2 ).

Example

Family income (X2) can be a positive moderator of the relationship between family size (X1) and credit card usage (Y). Then expected change in credit card usage based on family size (

𝛽

1 ) might be lower for families with low incomes and high in high incomes.

Without the moderator effect we are assuming that family size have a constant effect on credit card usage.

Adding Moderator Effect

The idea comes from observing a self moderator effect. If a variable has a moderator effect onto itself then we would assume a nonlinear (second degree) relationship with the dependent variable. Thus if there is a moderator effect add X 1 X 2 as an independent variable to regression equation. But we will return back to this!!!

Assumption: Homoscedasticity

Constant variance of the error terms.

•

Heteroscedasticity (cts.) in residuals within variables

• Heteroscedasticity (cts.)

Use: Levene Test.

Levene Test: Tests the equality of variance.

Levene's test works by testing the null hypothesis that the variances of the group are the same. The output probability is the probability that at least one of the samples in the test has a significantly different variance. If this is greater than a selected percentage (usually 5%) then it is considered too great to be able to usefully apply parametric tests.

Variances In SPSS it is reported.

In R: In «lawstat» library use levene.test() function.

Use F test for more than 2 groups…

Residuals vs Fitted 26 2 25 0 5 10 15 20 Fitted values lm(leslie_salt[, 1] ~ leslie_salt[, 4] + leslie_salt[, 5] + leslie_salt[, 6 ...

Assumptions

•

Independence of the error terms.

Check the coordinates!!!

• Independence of Error Terms

Use: Durbin-Watson The value of the Durbin-Watson statistic ranges from 0 to 4. As a general rule of thumb, the residuals are uncorrelated is the Durbin Watson statistic is approximately 2. A value close to 0 indicates strong positive correlation, while a value of 4 indicates strong negative correlation.

•

In SPSS Durbin Watson is reported.

•

In R under «lmtest» library use dwtest() dwtest(formula, order.by = NULL, alternative = c("greater", "two.sided", "less"), iterations = 15, exact = NULL, tol = 1e-10, data = list()) For our regression model.

> dwtest(model) Durbin-Watson test data: model DW = 2.3762, p-value = 0.7783 alternative hypothesis: true autocorrelation is greater than 0

•

Assumptions

Normality of the error term distribution.

Normal Q-Q 26 2 10 -2 -1 0 1 2 Theoretical Quantiles lm(leslie_salt[, 1] ~ leslie_salt[, 4] + leslie_salt[, 6] + leslie_salt[, 7 ...

qqPlot(model) -2 -1 0 t Quantiles 1 2

Diagonistics

Call: lm(formula = leslie_salt[, 1] ~ leslie_salt[, 4] + leslie_salt[, 5] + leslie_salt[, 6]) Residuals: Min 1Q Median 3Q Max -9.6076 -3.2506 -0.0281 2.8770 20.2776 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 21.2787636 2.9203157 7.286 7.75e-08 *** leslie_salt[, 4] 0.5614588 0.2515472 2.232 0.034107 * leslie_salt[, 5] -0.0005871 0.0004460 -1.316 0.199129 leslie_salt[, 6] 0.1836824 0.0421712 4.356 0.000172 *** -- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 5.559 on 27 degrees of freedom Multiple R-squared: 0.5327, Adjusted R-squared: 0.4807 F-statistic: 10.26 on 3 and 27 DF, p-value: 0.000111

Identifying Influential Observations •

observations that lie outside the general patterns of the data set

•

observations that strongly influence regression results Types of Influential Observations 1. Outliers – observations that have large residuals (based on dependent variables) 2. Leverage points – observations that are distinct from the remaining observations based on their independent variable values.

3. Influential observations – including all observations that have a disproportionate effect on the regression results.

Outliers •

Typical boxplot test.

•

In «car» library  2 outlierTest(model) rstudent unadjusted p-value 3.704906 0.0010527 Bonferonni p 0.032634

Leverage

An observation with an extreme value on a predictor variable is called a point with high leverage. Leverage is a measure of how far an IV deviates from its mean. These leverage points can have an unusually large effect on the estimate of regression coefficients. We hope to see very few (if any) points in the plot representing high values of leverage. High leverage can also point toward outliers, which are defined as observations with large residuals in regression. You should say something about the number of cases that appear to represent high leverage.

Leverage: Cut off point : 2 𝑝+1 𝑛 p: # of independent variables n: # of observations

Leverage Plots

-0.5

0.0

0.5

leslie_salt[, 4] | others 1.0

-0.5

0.0

0.5

leslie_salt[, 6] | others 1.0

-0.6

-0.2

0.2 0.4

leslie_salt[, 7] | others -0.2 0.0

0.2

0.4

leslie_salt[, 8] | others 0.6

0 5 10 15 Index 20 25 30

Cook’s Distance: Cut off point: 4 𝑛 p: # of independent variables n: # of observations

R-Code

# Influential Observations # added variable plots av.Plots(model) # Cook's D plot # identify D values > 4/(n-k-1) cutoff <- 4/((nrow(leslie_salt) length(model$coefficients)-2)) plot(fit, which=4, cook.levels=cutoff) # Influence Plot influencePlot(model, id.method="identify", main="Influence Plot", sub="Circle size is proportial to Cook's Distance" )

Influence Plot

2 26 5 30 21 10 0.1

0.2

4 0.3

0.4

Hat-Values Circle size is proportial to Cook's Distance 9

Residuals vs Leverage 2 Cook's distance 0.0

0.1

0.2

Leverage lm(leslie_salt[, 1] ~ leslie_salt[, 4] + leslie_salt[, 6] + leslie_salt[, 7 ...

0.3

4 0.4

9 1 0.5

Cook’s Distance: Cut off point: 4 𝑛 p: # of independent variables n: # of observations 0.5

Assessing Multicollinearity*****

A key issue in interpreting the regression variate is the correlation among the independent variables.

Our task in a regression analysis includes the following: 1. Assess the degree of multicollinearity 2. Determine its impact on results 3. Apply the necessary remedies if needed

Assess the degree of multicollinearity •

The simplest and most obvious way: Identifying collinearity in correlation matrix. Check for correlation >90%.

•

A direct measure of multicollinearity is tolerance (1/VIF). • The amount of variability of the selected independent variable not explained by the other independent variables. Computation: • Take each independent variable. Assume it as the dependent variable. Compute adjusted R 2 . • Tolerance is then 1-R 2 . • For example if other variables explain 25% of an independent variable then tolerence of this variable is 75%. Tolerence should be more than 10% > 1/vif(model) leslie_salt[, 4] leslie_salt[, 6] leslie_salt[, 7] leslie_salt[, 8] 0.8081325 0.9959058 0.7650806 0.7715437

Further…

•

see page http://www.statmethods.net/stats/rdiagnostics.html

diagonistic tests with R for

Partial Correlation •

A partial correlation coefficient is a way of expressing the unique relationship between the criterion and a predictor. Partial correlation represents the correlation between the criterion and a predictor after common variance with other predictors has been removed from both the criterion and the predictor of interest. t.values <- model$coeff / sqrt(diag(vcov(model))) partcorr <- sqrt((t.values^2) / ((t.values^2) + model$df.residual)) partcorr ***************************************************** leslie_salt[, 4] leslie_salt[, 6] leslie_salt[, 7] leslie_salt[, 8] 0.6562662 0.8043296 0.6043579 0.5740840

Part (Semi-partial) Correlation

• A semipartial correlation coefficient represents the correlation between the criterion and a predictor that has been residualized with respect to all other predictors in the equation. Note that the criterion remains unaltered in the semipartial. Only the predictor is residualized. After removing variance that the predictor has in common with other predictors, the semipartial expresses the correlation between the residualized predictor and the unaltered criterion. An important advantage of the semipartial is that the denominator of the coefficient (the total variance of the criterion, Y) remains the same no matter which predictor is being examined. This makes the semipartial very interpretable. The square of the semipartial can be interpreted as the proportion of the criterion variance associated uniquely with the predictor. It is also possible to use the semipartial to fully deconstruct the variance components in a regression analysis.

Project (Step1):

Go to web page: http://luna.cas.usf.edu/~mbrannic/files/regression/Partial.html

Replicate the results there using a dataset of your own. Be creative in problem formulation. Data may be imaginary. Use at least 5 independent variables.

Comparing Regression Models • •

In multiple regression the hardest problem is deciding on which variables to enter into equation even after checking assumption such as multicollinearity. Adjusted

𝑅

2 is not a proper way of model comparison.

•

Next we learn a better way.

Stepwise Regression •

Start with the most basic model. Pick your favourite independent variable and construct the model. Test it.

Remember correlation matrix (price in logs)

PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE PRICE

100.00%

COUNTY

-18.22%

SIZE

-23.97%

ELEVATION

35.18%

SEWER

-39.12%

DATE

59.47%

FLOOD

-32.31%

DISTANCE

9.33% -18.22% -23.97% 35.18% 100.00% -33.94% 47.52% -33.94% 100.00% -20.95% 47.52% -20.95% 100.00% -5.00% 5.34% -35.94% -36.98% -34.95% -5.65% -55.18% 10.89% -37.31% -74.22% 55.69% -36.25% -39.12% 59.47% -32.31% 9.33% -5.00% -36.98% -55.18% -74.22% 5.34% -34.95% 10.89% 55.69% -35.94% -5.65% -37.31% -36.25% 100.00% -15.15% -11.31% -15.87% -15.15% 100.00% 1.54% 4.44% -11.31% 1.54% 100.00% 42.33% -15.87% 4.44% 42.33% 100.00%

𝑝𝑟𝑖𝑐𝑒 = 𝛽

0

+ 𝛽

1

𝑑𝑎𝑡𝑒

𝑝𝑟𝑖𝑐𝑒 = 𝛽

0

+ 𝛽

1

𝑑𝑎𝑡𝑒

Call: lm(formula = leslie_salt[, 1] ~ leslie_salt[, 6]) Our focus is the improvement in RSS. So we need residual sum of squares. But it is not given in the report directly (given in SPSS).

Residuals: Min 1Q Median 3Q Max -1.12046 -0.34364 0.04853 0.39719 1.00081 > anova(m1) Analysis of Variance Table Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.322336 0.269975 12.306 4.9e-13 *** leslie_salt[, 6] 0.018124 0.004257 4.257 0.000198 *** -- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Response: leslie_salt[, 1] Df Sum Sq Mean Sq F value Pr(>F) leslie_salt[, 6] 1 5.9282 5.9282 18.124 0.0001982 *** Residuals 29 9.4858 0.3271 -- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.5719 on 29 degrees of freedom Multiple R-squared: 0.3846, Adjusted R-squared: 0.3634 F-statistic: 18.12 on 1 and 29 DF, p-value: 0.0001982

•

Now lets add another variable say SEWER and assume we have done all testing Call: lm(formula = leslie_salt[, 1] ~ leslie_salt[, 6] + leslie_salt[, 5]) Residuals: Min 1Q Median 3Q Max -1.21681 -0.21980 0.08597 0.29875 0.81520 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 3.442e+00 2.442e-01 14.093 3.07e-14 *** leslie_salt[, 6] 1.643e-02 3.841e-03 4.278 0.000199 *** leslie_salt[, 5] -1.105e-04 3.797e-05 -2.910 0.007013 ** -- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 0.51 on 28 degrees of freedom Multiple R-squared: 0.5275, Adjusted R-squared: 0.4937 F-statistic: 15.63 on 2 and 28 DF, p-value: 2.766e-05 Analysis of Variance Table Response: leslie_salt[, 1] Df Sum Sq Mean Sq F value Pr(>F) leslie_salt[, 6] 1 5.9282 5.9282 22.7903 5.146e-05 *** leslie_salt[, 5] 1 2.2024 2.2024 8.4671 0.007013 ** Residuals 28 7.2833 0.2601 ---

How much improvement do we have?

Our aim is to check whether the imrovement in RSS is statistically significant or not.

Define

𝐹

𝑖𝑚𝑝𝑟𝑜𝑣𝑒𝑚𝑒𝑛𝑡

= 𝑅𝑆𝑆

𝑜𝑙𝑑

− 𝑅𝑆𝑆 Δ 𝑖𝑛 𝑖𝑛𝑑. 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠 𝑅𝑆𝑆 𝑑. 𝑓.

𝑜𝑙𝑑 𝑜𝑙𝑑 𝑛𝑒𝑤 Numerator measures average improvement as we add a new variable (we may add a bunch of new variables) and scales the improvement with respect to original model.

The degrees of freedom of the statistic is (1,degrees of freedom of old model)

In our case 𝑅𝑆𝑆

𝑜𝑙𝑑

𝑅𝑆𝑆

𝑛𝑒𝑤

𝑑𝑓

𝑜𝑙𝑑

= 9.4858

= 7.2833

= 29 𝐹 = 2.2025

1 9.4858

29 = 6.733487

𝐹

𝑐𝑟𝑖𝑡

1,29 = 4.182964

So, new model is superior, the improvement is statistically significant.

Back to moderator effect.

To test the moderator effect we use,

𝑦 = 𝛽

0

+ 𝛽

1

𝑋

1 as the simple model and

𝑦 = 𝛽

0

+ 𝛽

1

𝑋

1

+ 𝛽

2

𝑋

1

𝑋

2 as the extended model and then decide accordingly.

Mediation

Project (Step2,3 and 4): •

Find the best regression equation for your Project.

•

Test moderator effects

•

Test mediation effects.