Interpretation in Multiple Regression

1 Interpretation in Multiple and Adjusted of parameter combinations of parameter estimates variance-covariance matrix standard errors of combinations standard error for the meanWe will use the final model from last time to illustrate these concepts. Summaries of themodel - least squares estimates with standard errors given below in parentheses: logit proportion log duration I .14 = with 44 degrees of freedomR-squared = :TheR-squaredvaluemeansthat61% ,theadjustedR-squaredisoftenusedtosummar izethefitasit takes into account the the number of variables in the model. Adjusted R-squared = 1 - Mean Square Error /Total Mean SquarewhereMeanSquareErroris 2fromtheregressionmodelandtheTotalmeansq uareisthesamplevarianceoftheresponse(sY2 2isagoodestimateifalltheregressioncoeffi cients are 0).

2 For this example,Adjusted R-squared = 1 - ^2/ = :theinterceptinamultipleregressionmodeli sthemeanfortheresponsewhenall of the explanatory variables take on the value ,thismeansthatthedummyvariableI=0(code=1 ,whichwasthequeenbumblebees)andlog(durat ion)=0, ,withvisitsof1second,weare95%confidentth atthemeanlogit(proportionofpollenremoved )isbetween is based on 44 degrees of freedom; qt(.975, 44).Toconvertbacktotheoriginalunits, (p)=log(p/(1-p)),thenp=exp(x)/(1+exp(x)) . ,weare95%confidentthatthemeanproportiono fpollenremoved is between and [exp( )/(1 + exp( )) to exp( )/(1 + exp( ))] Note: while this is the Interpretation of the intercept, we are ,keepingallothervariablesheldfixedisimpo ssible( ,orthemodelwithdifferentslopes for queen and worker bees).

3 Forthisexample,wehavetheestimatedcoeffic ientoflog(duration) taken the log transformation of duration, the Interpretation of the coefficient is easierto understand by looking at a doubling of duration (review page 208 chapter 8).Adoublingofthedurationofvisitcorrespo ndstoa 1log(2)changeinthemeanlogit(proportionof pollenremoved) *log(2)= (2).Soa95%confidenceintervalforthechange inthemeanlogit(proportionpollenremoved) is not possible. Dummy variable coefficients:A1unitchangeforadummyvariab leimpliesgoingfromlevel0tolevel1,sotheth einterpretationofthedummyvariablecoeffic ientistheamountbywhichthemeanlogit(propo rtion)forworkerbeesexceedsthemeanlogit(p roportion) for the amount is to (this is the case for parallel Regression lines; if westillhadtheinteractionvariablewecouldn otmakethisstatement,sincetheinteractiono fthe dummy*log(duration) cannot be held constant).

4 Inthemodelderivation,wesaidthattheinterc eptplusthedummyvariablecoefficientcorres pondedtotheinterceptfortheworkerbees, +. ,let'sfindaconfidenceintervalfor 0 + 2. To do this we need to find the standard error for a linear combination. Linear Combination of ParametersTofindthevariance(andthenstand arddeviation)oftheestimatorof 0+ 2weneedtotakeintoaccounttheindividualvar iancesplushowtheestimateswillvarytogethe rfromsampletosample(theircovariance). (recallthecorrelationisthecovariancedivi dedbytheproductofthestandarddeviations, , ,checkoff the box to get the estimated correlation matrix of the coefficients. Correlation of Coefficients: (Intercept) ---- I Thecorrelationbetweensomethinganditselfi sone, (b0,b1)isthesameasthecorrelationof(b1,b0 )thetableonlyincludes the elements below the diagonal.)

5 Theestimatedcovariancematrixissymmetric( justlikethecorrelationmatrix).Thediagona lelementsarethecovariancebetween iand iwhicharethevariances,orthesquare of the standard errors:Covariance Matrix of the Parameter Estimates coefficient (Intercept) I (Intercept) I covariance between the intercept and the dummy variable I coefficient is estimated asthe (correlation between the intercept and the coefficient for I ) * (SE(intercept))( SE(ofthe coefficient for I) ) or * * = standard error for the estimate of 0 + 2 is the sqrt( ^2 + ^2 + 2* ) = Thus a 95% Confidence interval for the intercept for worker bees is , Transforming back to the original units, we are 95% confident that the mean proportionof pollen removed by worker bees for visits of 1 second is between to Error of the Mean.

6 The same approach is used to find the standard error of the mean for a visit of a givenduration. the estimate of the mean of logit for a visit of 2 seconds for a queenbumble bee is * 1 + * log(2) + * 0 . The values C0 = 1, C1=log(2), C2=0are the multipliers Ci in the linear combination i 0p iCiwhich gives us the estimate of the linear combination (the mean in this case) The variance of the mean at this point is found by i 0p j 0pcov !"i,!"j#CiCjwhich in this case simplifies to var !"0#$1%var !"1#log 2#2%2$cov !"0,!"1#$1$log 2#& From the SE(mean) we can get the SE(prediction)SE prediction Y'X(x))(*Xhat+2, , of visit (seconds) pollen removedcode=1 Queenscode=2 WorkersEstimated Mean for QueensEstimated Mean for Workers95% Prediction Intervals Queens95% Prediction Intervals WorkersProportion of Pollen Removed for Queen Bumblebees and Worker Honeybees