Transcription of 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). 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(.)
2 975, 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).Forthisexample,wehavetheestimatedc oefficientoflog(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).
3 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. 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:The same approach is used to find the standard error of the mean for a visit of a givenduration.)
4 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
