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Lecture 6: Multiple Linear Regression Adjusted Variable Plots. BMTRY 701 Biostatistical Methods II. Graphical Displays in MLR. No more one simple scatterplot: need to look at multiple pairs of variables “pairs” in R. - PowerPoint PPT Presentation
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Lecture 6:Multiple Linear RegressionAdjusted Variable Plots
BMTRY 701Biostatistical Methods II
Graphical Displays in MLR
No more one simple scatterplot: need to look at multiple pairs of variables
“pairs” in R. but, we can’t look at all covariates in regards to
the way they enter the model solution: adjusted variable plot aka: partial regression plot
Adjusted Variable Plots
Adjusted variable plots are useful in linear regression for outlier detection and for qualitative evaluation of the fit of a model.
With two covariates: Shows the association between X and Y adjusted for another variable, Z.
With more than two covariates: Shows the association between X and Y adjusted for many other covariates
In our example, association between logLOS and number of nurses, adjusted for number of beds
Approach
Assume we want to look at the association of Y and X, adjusted for Z
Step 1: Regress Y on X and save residuals (res.xy)
Step 2: Regress Z on X and save residuals (res.xz)
Step 3: plot res.xy versus res.xz Optional step 4:
• perform regression of res.xy on res.xz• compare slope to that of MLR of Y on X and Z
MPV: section 4.2.4
SENIC
INFRISK
0 200 400 600 800
23
45
67
8
020
040
060
080
0
BEDS
2 3 4 5 6 7 8 2.0 2.2 2.4 2.6 2.8 3.0
2.0
2.2
2.4
2.6
2.8
3.0
logLOS
R
pairs(~INFRISK+BEDS+logLOS, data=data, pch=16)
# adjusted variable plot approach# look at the association between INFRISK and logLOS, # adjusting for BEDS
reg.xy <- lm(logLOS ~ BEDS, data=data)res.xy <- reg.xy$residuals
reg.xz <- lm(INFRISK ~ BEDS, data=data)res.xz <- reg.xz$residuals
plot(res.xz, res.xy, pch=16)reg.res <- lm(res.xy ~ res.xz)abline(reg.res, lwd=2)reg.infrisk.beds <- lm(logLOS ~ BEDS + INFRISK, data=data)
-2 -1 0 1 2 3
-0.2
0.0
0.2
0.4
0.6
res.xz
res.
xy
Why is this important or interesting?
It shows us the ‘adjusted’ relationship it can help us determine if
• it is an important variable (at all)• if another form of X is more appropriate• if the correlation is high vs. low after adjustment• we need to/want to adjust for this variable
It also informs us about why a variable ‘loses’ significance
Most important: check for non-linearity Example: logLOS ~ NURSE
What about BEDS and NURSE?
# why NURSE is not associated, after adjustment for BEDS?
reg.nurse <- lm(logLOS ~ NURSE, data=data)reg.nurse.beds <- lm(logLOS ~ NURSE + BEDS, data=data)
reg.xy <- lm(logLOS ~ BEDS, data=data)res.xy <- reg.xy$residuals
reg.xz <- lm(NURSE ~ BEDS, data=data)res.xz <- reg.xz$residuals
plot(res.xz, res.xy, pch=16)reg.res <- lm(res.xy ~ res.xz)abline(reg.res, lwd=2)
-200 -100 0 100 200
-0.2
0.0
0.2
0.4
0.6
res.xz
res.
xy
What about the other way around?
######################## what about the other way? what about why BEDS is # assoc after adjustment for NURSE?
reg.xy <- lm(logLOS ~ NURSE, data=data)res.xy <- reg.xy$residuals
reg.xz <- lm(BEDS ~ NURSE, data=data)res.xz <- reg.xz$residuals
plot(res.xz, res.xy, pch=16)reg.res <- lm(res.xy ~ res.xz)abline(reg.res, lwd=2)reg.nurse.beds <- lm(logLOS ~ NURSE + BEDS, data=data)
-200 -100 0 100 200 300
-0.2
0.0
0.2
0.4
0.6
res.xz
res.
xy
Interpretation in MLR
“Adjusted for” “Controlled for “ “Holding all else constant”
In MLR, you need to include one of these phrases (or something like one of them) when interpreting a regression coefficient
LOS ~ INFRISK + BEDS > reg.infrisk.beds <- lm(LOS ~ BEDS + INFRISK, data=data)> summary(reg.infrisk.beds)
Call:lm(formula = LOS ~ BEDS + INFRISK, data = data)
Residuals: Min 1Q Median 3Q Max -2.8624 -0.9904 -0.1996 0.6671 8.4219
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.2703521 0.5038751 12.444 < 2e-16 ***BEDS 0.0024747 0.0008236 3.005 0.00329 ** INFRISK 0.6323812 0.1184476 5.339 5.08e-07 ***---
Hard to interpret with so many decimal places!
> data$beds100 <- data$BEDS/100> reg.infrisk.beds <- lm(LOS ~ beds100 + INFRISK, data=data)> summary(reg.infrisk.beds)
Call:lm(formula = LOS ~ beds100 + INFRISK, data = data)
Residuals: Min 1Q Median 3Q Max -2.8624 -0.9904 -0.1996 0.6671 8.4219
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 6.27035 0.50388 12.444 < 2e-16 ***beds100 0.24747 0.08236 3.005 0.00329 ** INFRISK 0.63238 0.11845 5.339 5.08e-07 ***---
logLOS ~ INFRISK + BEDS
> reg.infrisk.beds <- lm(logLOS ~ BEDS + INFRISK, data=data)> summary(reg.infrisk.beds)
Call:lm(formula = logLOS ~ BEDS + INFRISK, data = data)
Residuals: Min 1Q Median 3Q Max -0.314377 -0.079979 -0.008026 0.072108 0.580675
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.926e+00 4.611e-02 41.767 < 2e-16 ***BEDS 2.407e-04 7.538e-05 3.194 0.00183 ** INFRISK 6.048e-02 1.084e-02 5.579 1.75e-07 ***---
Hard to interpret with so many decimal places!
> data$beds100 <- data$BEDS/100> reg.infrisk.beds100 <- lm(logLOS ~ beds100 + INFRISK, data=data)> summary(reg.infrisk.beds100)
Call:lm(formula = logLOS ~ beds100 + INFRISK, data = data)
Residuals: Min 1Q Median 3Q Max -0.314377 -0.079979 -0.008026 0.072108 0.580675
Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.926040 0.046114 41.767 < 2e-16 ***beds100 0.024075 0.007538 3.194 0.00183 ** INFRISK 0.060477 0.010840 5.579 1.75e-07 ***---Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.1435 on 110 degrees of freedomMultiple R-squared: 0.3612, Adjusted R-squared: 0.3496 F-statistic: 31.1 on 2 and 110 DF, p-value: 1.971e-11
How to interpret?
Pick two values of BEDS• e.g. 100 to 200• e.g. 400 to 500
Estimate the difference in logLOS for each value
What do we plug in for INFRISK?
INFRISKbedsINFRISKbedsSLO
*060.0100*024.093.1
ˆ100ˆˆˆlog 210
How to interpret? Remember that our inferences are “holding all else
constant” To compare two hospitals with the same INFRISK, it
doesn’t matter what you put in (as long as it is the same)
024.01*024.02*024.0
)*060.01*024.093.1()*060.02*024.093.1(]1100|ˆ[log]2100|ˆ[log
*060.02*024.093.1
ˆ100ˆˆ]2100|ˆ[log
*060.01*024.093.1
ˆ100ˆˆ]1100|ˆ[log
210
210
INFRISKINFRISKbedsSLObedsSLO
INFRISKINFRISKbedsbedsSLO
INFRISKINFRISKbedsbedsSLO
How to interpret?
02.1)024.0exp()1(ˆ)2(ˆ
)1(ˆ)2(ˆlogexp))1(ˆlog)2(ˆexp(log
SLOSLO
SLOSLOSLOSLO
Comparing two hospitals whose number of beds differ by 100 andassuming the same infection risk in the two hospitals is the same, theratio of average LOS in the two hospitals is 1.02 with the hospital with more beds having the longer stay.
difference of 400 beds?
10.1)024.0*4exp()1(ˆ)5(ˆ
)1(ˆ)5(ˆlogexp))1(ˆlog)5(ˆexp(log
SLOSLO
SLOSLOSLOSLO
When outcome is log transformed
interpretation of coefficients must be made as RATIOS instead of DIFFERENCES
Need to exponentiate the coefficient. its interpretation is the ratio for a one-unit
difference in the predictor.
Why differences do not work
Consider comparing two hospitals with 400 and 300 beds:
)*06.093.1exp(*0263.0)075.1101.1)(*06.093.1exp(
))3*024.0exp()4*024.0)(exp(*06.093.1exp()*06.0exp(*)3*024.0exp(*)93.1exp()*06.0exp(*)4*024.0exp(*)93.1exp(]3100|ˆ[]4100|ˆ[
)*06.0exp(*)3*024.0exp(*)93.1exp()*06.03*024.093.1exp(]3100|ˆ[
*060.03*024.093.1
ˆ100ˆˆ]3100|ˆ[log
)*06.0exp(*)4*024.0exp(*)93.1exp()*06.04*024.093.1exp(]4100|ˆ[
*060.04*024.093.1
ˆ100ˆˆ]4100|ˆ[log
210
210
INFRISKINFRISKINFRISK
INFRISKINFRISKbedsSLObedsSLO
INFRISKINFRISKbedsSOL
INFRISKINFRISKbedsbedsSLO
INFRISKINFRISKbedsSOL
INFRISKINFRISKbedsbedsSLO
Why differences do not work
Consider comparing two hospitals with 800 and 700 beds:
)*06.093.1exp(*0291.0)183.1212.1)(*06.093.1exp(
))7*024.0exp()8*024.0)(exp(*06.093.1exp()*06.0exp(*)7*024.0exp(*)93.1exp()*06.0exp(*)8*024.0exp(*)93.1exp(]7100|ˆ[]8100|ˆ[
)*06.0exp(*)7*024.0exp(*)93.1exp()*06.07*024.093.1exp(]7100|ˆ[
*060.07*024.093.1
ˆ100ˆˆ]7100|ˆ[log
)*06.0exp(*)8*024.0exp(*)93.1exp()*06.08*024.093.1exp(]8100|ˆ[
*060.08*024.093.1
ˆ100ˆˆ]8100|ˆ[log
210
210
INFRISKINFRISKINFRISK
INFRISKINFRISKbedsSLObedsSLO
INFRISKINFRISKbedsSOL
INFRISKINFRISKbedsbedsSLO
INFRISKINFRISKbedsSOL
INFRISKINFRISKbedsbedsSLO
Results in the log scale
0 2 4 6 8
2.0
2.2
2.4
2.6
2.8
3.0
data$beds100
data
$log
LOS
INFRISK=2INFRISK=5
Results on the “linear” scale: not huge differences
0 2 4 6 8
810
1214
1618
20
data$beds100
data
$LO
SINFRISK=2INFRISK=5
Differences can be seen on a larger scale plot
0 10 20 30 40 50
1015
2025
30
data$beds100
data
$LO
S
INFRISK=2INFRISK=5