Multiple Regression

Objectives

• 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.

Y’ X1

X2

X3

Sample Problem (Leslie Salt Property):Finding Fair Price of a LandVariable Name Description

PRICE Sale price in $000 per acre

COUNTY San Mateo=0, Santa Clara=1

SIZE Size of the property in acres

ELEVATION Average elevation in feet above sea level

SEWER Distance (in feet) to nearest sewer connection

DATE Date of sale counting backward from current time (in months)

FLOOD Subject to flooding by tidal action =1; otherwise=0

DISTANCE Distance in miles from Leslie property

PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE4.5 1 138.4 10 3000 -103 0 0.3

10.6 1 52 4 0 -103 0 2.51.7 0 16.1 0 2640 -98 1 10.3

5 0 1695.2 1 3500 -93 0 145 0 845 1 1000 -92 1 14

3.3 1 6.9 2 10000 -86 0 05.7 1 105.9 4 0 -68 0 06.2 1 56.6 4 0 -64 0 0

19.4 1 51.4 20 1300 -63 0 1.23.2 1 22.1 0 6000 -62 0 04.7 1 22.1 0 6000 -61 0 06.9 1 27.7 3 4500 -60 0 08.1 1 18.6 5 5000 -59 0 0.5

11.6 1 69.9 8 0 -59 0 4.419.3 1 145.7 10 0 -59 0 4.211.7 1 77.2 9 0 -59 0 4.513.3 1 26.2 8 0 -59 0 4.715.1 1 102.3 6 0 -59 0 4.912.4 1 49.5 11 0 -59 0 4.615.3 1 12.2 8 0 -59 0 512.2 0 320.6 0 4000 -54 0 16.518.1 1 9.9 5 0 -54 0 5.216.8 1 15.3 2 0 -53 0 5.5

5.9 0 55.2 0 1320 -49 1 11.94 0 116.2 2 900 -45 1 5.5

37.2 0 15 5 0 -39 0 7.218.2 0 23.4 5 4420 -39 0 5.515.1 0 132.8 2 2640 -35 0 10.222.9 0 12 5 3400 -16 0 5.515.2 0 67 2 900 -5 1 5.521.9 0 30.8 2 900 -4 0 5.5

Histogram of leslie_salt[, 1]

leslie_salt[, 1]

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EPRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCE

PRICE 100.00% -18.22% -23.97% 35.18% -39.12% 59.47% -32.31% 9.33%COUNTY -18.22% 100.00% -33.94% 47.52% -5.00% -36.98% -55.18% -74.22%SIZE -23.97% -33.94% 100.00% -20.95% 5.34% -34.95% 10.89% 55.69%ELEVATION 35.18% 47.52% -20.95% 100.00% -35.94% -5.65% -37.31% -36.25%SEWER -39.12% -5.00% 5.34% -35.94% 100.00% -15.15% -11.31% -15.87%DATE 59.47% -36.98% -34.95% -5.65% -15.15% 100.00% 1.54% 4.44%FLOOD -32.31% -55.18% 10.89% -37.31% -11.31% 1.54% 100.00% 42.33%DISTANCE 9.33% -74.22% 55.69% -36.25% -15.87% 4.44% 42.33% 100.00%

PRICE

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PRICE COUNTY SIZE ELEVATION SEWER DATE FLOOD DISTANCEPRICE 100.00% -18.22% -23.97% 35.18% -39.12% 59.47% -32.31% 9.33%COUNTY -18.22% 100.00% -33.94% 47.52% -5.00% -36.98% -55.18% -74.22%SIZE -23.97% -33.94% 100.00% -20.95% 5.34% -34.95% 10.89% 55.69%ELEVATION 35.18% 47.52% -20.95% 100.00% -35.94% -5.65% -37.31% -36.25%SEWER -39.12% -5.00% 5.34% -35.94% 100.00% -15.15% -11.31% -15.87%DATE 59.47% -36.98% -34.95% -5.65% -15.15% 100.00% 1.54% 4.44%FLOOD -32.31% -55.18% 10.89% -37.31% -11.31% 1.54% 100.00% 42.33%DISTANCE 9.33% -74.22% 55.69% -36.25% -15.87% 4.44% 42.33% 100.00%

log (𝑃𝑅𝐼𝐶𝐸 )=𝑏0+𝑏1𝐸𝐿𝐸𝑉𝐴𝑇𝐼𝑂𝑁 +𝑏2𝑆𝐸𝑊𝐸𝑅+𝑏3𝐷𝐴𝑇𝐸+𝑏4𝐹𝐿𝑂𝑂𝐷+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 freedomMultiple 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.

𝑋 1

𝑦

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 (X1) changes across values of the moderator variable (X2).

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 () 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 X1X2 as an independent variable to regression equation. But we will return back to this!!!

Assumption:HomoscedasticityConstant 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…

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Assumptions

• Independence of the error terms.

Check the coordinates!!!

Independence of Error Terms• Use: Durbin-WatsonThe 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.

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Theoretical Quantiles

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

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DiagonisticsCall: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 freedomMultiple 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 Observations1. 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

outlierTest(model) rstudent unadjusted p-value Bonferonni p

2 3.704906 0.0010527 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 : p: # of independent variablesn: # of observations

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Cook’s Distance:Cut off point: p: # of independent variablesn: # 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" )

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Circle size is proportial to Cook's DistanceHat-Values

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

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Cook’s Distance:Cut off point: p: # of independent variablesn: # of observations

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 multicollinearity2. Determine its impact on results3. 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

R2. • Tolerance is then 1-R2.

• 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 for diagonistic tests with R

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 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 DISTANCEPRICE 100.00% -18.22% -23.97% 35.18% -39.12% 59.47% -32.31% 9.33%COUNTY -18.22% 100.00% -33.94% 47.52% -5.00% -36.98% -55.18% -74.22%SIZE -23.97% -33.94% 100.00% -20.95% 5.34% -34.95% 10.89% 55.69%ELEVATION 35.18% 47.52% -20.95% 100.00% -35.94% -5.65% -37.31% -36.25%SEWER -39.12% -5.00% 5.34% -35.94% 100.00% -15.15% -11.31% -15.87%DATE 59.47% -36.98% -34.95% -5.65% -15.15% 100.00% 1.54% 4.44%FLOOD -32.31% -55.18% 10.89% -37.31% -11.31% 1.54% 100.00% 42.33%DISTANCE 9.33% -74.22% 55.69% -36.25% -15.87% 4.44% 42.33% 100.00%

𝑝𝑟𝑖𝑐𝑒=𝛽0+𝛽1𝑑𝑎𝑡𝑒

𝑝𝑟𝑖𝑐𝑒=𝛽0+𝛽1𝑑𝑎𝑡𝑒

Call:lm(formula = leslie_salt[, 1] ~ leslie_salt[, 6])

Residuals: Min 1Q Median 3Q Max -1.12046 -0.34364 0.04853 0.39719 1.00081

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

Residual standard error: 0.5719 on 29 degrees of freedomMultiple R-squared: 0.3846, Adjusted R-squared: 0.3634 F-statistic: 18.12 on 1 and 29 DF, p-value: 0.0001982

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).

> anova(m1)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 18.124 0.0001982 ***Residuals 29 9.4858 0.3271 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

• 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 freedomMultiple 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

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

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

Back to moderator effect.

To test the moderator effect we use,

as the simple model and

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.