Lecture 11Multicollinearity

BMTRY 701Biostatistical Methods II

Multicollinearity Introduction

Some common questions we ask in MLR• what is the relative importance of the effects of the

different covariates?• what is the magnitude of effect of a given covariate on

the response?• can any covariate be dropped from the model

because it has little effect or no effect on the outcome?

• should any covariates not yet included in the model be considered for possible inclusion?

Easy answers?

If the candidate covariates are uncorrelated with one another: yes, these are simple questions

If the candidate covariates are correlated with one another: no, these are not easy.

Most commonly:• observational studies have correlated covariates• we need to adjust for these when assessing relationships• “adjusting” for confounders

Experimental designs?• less problematic• patients are randomized in common designs• no confounding exists because factors are ‘balanced’ across

arms

Multicollinearity

Also called “intercorrelation” refers to the situation when the covariates are

related to each other and to the outcome of interest

like confounding, but a statistical terminology for it because of the effects it has on regression modeling

No Multicollinearity Example: Mouse experiment

Mouse Dose A Dose B Diet Tumor size

1 100 25 0 45

2 200 25 0 56

3 300 25 0 25

4 100 50 0 15

5 200 50 0 17

6 300 50 0 10

7 100 25 1 30

8 200 25 1 28

9 300 25 1 20

10 100 50 1 10

11 200 50 1 5

12 300 50 1 3

Linear modeling

Interested in seeing which factors influence tumor size in mice

Notice that the experiment is perfectly balanced. What does that mean?

Dose of Drug A on Tumor

> reg.a <- lm(Tumor.size ~ Dose.A, data=data)> summary(reg.a)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 32.50000 12.29041 2.644 0.0246 *Dose.A -0.05250 0.05689 -0.923 0.3779 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 16.09 on 10 degrees of freedomMultiple R-Squared: 0.07847, Adjusted R-squared: -0.01368 F-statistic: 0.8515 on 1 and 10 DF, p-value: 0.3779

>

Dose of Drug B on Tumor

> reg.b <- lm(Tumor.size ~ Dose.B, data=data)> summary(reg.b)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 58.0000 9.4956 6.108 0.000114 ***Dose.B -0.9600 0.2402 -3.996 0.002533 ** ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 10.4 on 10 degrees of freedomMultiple R-Squared: 0.6149, Adjusted R-squared: 0.5764 F-statistic: 15.97 on 1 and 10 DF, p-value: 0.002533

>

Diet on Tumor

> reg.diet <- lm(Tumor.size ~ Diet, data=data)> summary(reg.diet)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 28.000 6.296 4.448 0.00124 **Diet -12.000 8.903 -1.348 0.20745 ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 15.42 on 10 degrees of freedomMultiple R-Squared: 0.1537, Adjusted R-squared: 0.06911 F-statistic: 1.817 on 1 and 10 DF, p-value: 0.2075

All in the model together

> reg.all <- lm(Tumor.size ~ Dose.A + Dose.B + Diet, data=data)> summary(reg.all)

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 74.50000 8.72108 8.543 2.71e-05 ***Dose.A -0.05250 0.02591 -2.027 0.077264 . Dose.B -0.96000 0.16921 -5.673 0.000469 ***Diet -12.00000 4.23035 -2.837 0.021925 * ---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.327 on 8 degrees of freedomMultiple R-Squared: 0.8472, Adjusted R-squared: 0.7898 F-statistic: 14.78 on 3 and 8 DF, p-value: 0.001258

Correlation matrix of predictors and outcome

> cor(data[,-1]) Dose.A Dose.B Diet Tumor.sizeDose.A 1.0000000 0.0000000 0.0000000 -0.2801245Dose.B 0.0000000 1.0000000 0.0000000 -0.7841853Diet 0.0000000 0.0000000 1.0000000 -0.3920927Tumor.size -0.2801245 -0.7841853 -0.3920927 1.0000000>

Result

For perfectly balanced designs, adjusting does not affect the coefficients

However, it can affect the significance Why?

• residual sum of squares is affected• if you explain more of the variance in the outcome,

less is left to chance/error• when you adjust for another related factor, you will

likely improve the significance

The other extreme: perfect collinearity

Mouse Dose A Dose C Diet Tumor size1 100 100 0 452 200 300 0 563 300 500 0 254 100 100 0 155 200 300 0 176 300 500 0 107 100 100 1 308 200 300 1 289 300 500 1 2010 100 100 1 1011 200 300 1 512 300 500 1 3

The model has infinitely many solutions

Too much flexibility What happens? The fitting algorithm usually gives you some

indication of this• will not fit the model and gives an error• drops one of the predictors

“perfectly collinear” = “perfect confounding”

Effects of Multicollinearity

Most common result• two covariates are independently associated with Y in

simple linear regression models• in MLR model with both covariates, one or both is

insignificant• the magnitude of the regression coefficients is

attenuated• why?

recall the adjusted variable plot if the two are related, removing the systematic part of one

from Y may leave too little left to explain

Effects of Multicollinearity

Other situations• Neither is significant alone, but they are both

significant together (somewhat rare)• Both are significant alone and both retain signficance

in the model• The regression coefficient for one of the covariates

may change direction• Magnitude of coefficient may increase (in absolute

value) It is usually hard to predict exactly what will

happen when both are in the model

Implications in inference

the interpretation of a regression coefficient measuring the change in the expected value of Y when the covariate is increased while all other are held constant is not quite applicable

It may be conceptually feasible to think of ‘holding all constant’

but, practically, it may not be possible if the covariates are related.

Example: amount of rainfall and hours of sunshine

Implications in inference

multicollinearity tends to inflate the standard errors on the regression coefficients

when multicollinearity is present, you will see that coefficient of partial determination will have little increase with the addition of the collinear covariate

Predictions tend to be relatively unaffected for better or worse when a highly collinear covariate is added to the model.

Implications in Inference

Recall the interpretation of the t-statistics in MLR The represent the significance of a variable,

adjusting for all else in the model If two covariates are highly correlated, then both

are likely to end up insignificant Marginal nature of t-tests! ANOVA can be more useful due to conditional

nature of tables.

So, which is the ‘correct’ variable?

Almost impossible to tell Usually, people choose the one that is ‘more’

significant. but that does not mean it is the correct choice

• it could be the correct choice• it could be the one that is less associated

why might it be less associated? measurement issues

• the correct ‘culprit’ could be a variable that is related to the ones in the model but not in the model itself.

Example

Let’s look at our classic example of logLOS What variables are associated with logLOS? What variables have the potential to create

multicollinearity?

SENIC

> data <- read.csv("senicfull.csv")> data$logLOS <- log(data$LOS)> data$nurse2 <- data$NURSE^2> data$ms <- ifelse(data$MEDSCHL==2,0,data$MEDSCHL)> > data.cor <- data[,-1]> round(cor(data.cor),2) LOS AGE INFRISK CULT XRAY BEDS MEDSCHL REGION CENSUS NURSE FACS logLOS nurse2 msLOS 1.00 0.19 0.53 0.33 0.38 0.41 -0.30 -0.49 0.47 0.34 0.36 0.98 0.25 0.30AGE 0.19 1.00 0.00 -0.23 -0.02 -0.06 0.15 -0.02 -0.05 -0.08 -0.04 0.17 -0.04 -0.15INFRISK 0.53 0.00 1.00 0.56 0.45 0.36 -0.23 -0.19 0.38 0.39 0.41 0.55 0.26 0.23CULT 0.33 -0.23 0.56 1.00 0.42 0.14 -0.24 -0.31 0.14 0.20 0.19 0.35 0.15 0.24XRAY 0.38 -0.02 0.45 0.42 1.00 0.05 -0.09 -0.30 0.06 0.08 0.11 0.39 0.04 0.09BEDS 0.41 -0.06 0.36 0.14 0.05 1.00 -0.59 -0.11 0.98 0.92 0.79 0.42 0.86 0.59MEDSCHL -0.30 0.15 -0.23 -0.24 -0.09 -0.59 1.00 0.10 -0.61 -0.59 -0.52 -0.32 -0.56 -1.00REGION -0.49 -0.02 -0.19 -0.31 -0.30 -0.11 0.10 1.00 -0.15 -0.11 -0.21 -0.52 -0.07 -0.10CENSUS 0.47 -0.05 0.38 0.14 0.06 0.98 -0.61 -0.15 1.00 0.91 0.78 0.48 0.84 0.61NURSE 0.34 -0.08 0.39 0.20 0.08 0.92 -0.59 -0.11 0.91 1.00 0.78 0.37 0.95 0.59FACS 0.36 -0.04 0.41 0.19 0.11 0.79 -0.52 -0.21 0.78 0.78 1.00 0.38 0.66 0.52logLOS 0.98 0.17 0.55 0.35 0.39 0.42 -0.32 -0.52 0.48 0.37 0.38 1.00 0.28 0.32nurse2 0.25 -0.04 0.26 0.15 0.04 0.86 -0.56 -0.07 0.84 0.95 0.66 0.28 1.00 0.56ms 0.30 -0.15 0.23 0.24 0.09 0.59 -1.00 -0.10 0.61 0.59 0.52 0.32 0.56 1.00>

Let’s try an example with serious multicollinearity

To anticipate multicollinearity, ALWAYS good to look at scatterplots and correlation matrices of potential covariates

What covariates would give rise to a good example?

data.logLOS

0 200 600 0 200 400 600

2.0

2.2

2.4

2.6

020

060

0

data.BEDS

data.CENSUS

020

040

060

0

2.0 2.2 2.4 2.6

020

040

060

0

0 200 400 600

data.NURSE