The General Linear Model (GLM): A gentle introduction

1 Chapter9 The General Linear Model (GLM): A gentle Example with a single predictor s start with an example. Schizophrenics smoke a lot. They smoke be-tween two and three times more than the General population and about 50%more than those with other types of psychopathology (??). Obviously, expli-cating the nature of this relationship might provide insights into the etiology early type of research into this area compared the density of cholingergicnicotinic receptors (nAChR) in the brains of schizophrenics and controls (?).The data set Schizophrenia and nicotinic receptors shown in Table giveshypothetical data of such a study done in the past when analysis of post mortembrain specimens was the only way to examine this the moment, ignore the variables Age, Smoke and Cotinine and let usask the simple question of whether schizophrenics have more or fewer nicotinicreceptors in the brain area used in this study.

2 The operative word in the gen-eral Linear Model (GLM) is Linear . That word, of course, implies a straightline. Hence, mathematically we begin with the equation for a straight line. Instatisticalese, we write Y= 0+ 1X( )Read the predicted value of the a variable ( Y)equalsaconstantorintercept( 0)plus a weight or slope( 1)times the value of another variable (X). Let slook at the data first by plottingY(not Y)asafunctionofX,orintheexample,variable nAChR as a function of variable Schizophrenia (see Figure ).The purpose of a GLM is to fit a straight line through the points in Here is where the 0is the intercept for astraight line, , the value ofYwhenXis 0.

3 1is the slope of the line. When 1=0, then the predicted nAChR density for schizophrenics is the same EXAMPLE WITH A SINGLE PREDICTOR 9. THE General Linear Model (GLM): A GENTLEINTRODUCTIONT able : Data set on schizophrenia and brain density of nicotinic SzDummyCode Age Smoke Cotinine nAChR1No0 52? 56? 80? 84? ? ? ? ? ? ? ? ? ? 9. THE General Linear Model (GLM): A EXAMPLE WITH A SINGLE PREDICTOR : Number of nicotinic receptors (nAChR) as a function of for controls. As the slope deviates from 0, in either a positive or negativedirection, then there is more and more this point, you may rightly ask how one can have an intercept and a slopefor a variable that has values of No and Yes.

4 We will see the answer later,but for the time being let us create a numeric variable called SzDummyCodethat has the numerical value of 1 for Schizophrenia = Yes and 0 the GLM gives these estimates: 0= 1= ,for controls, the value ofXin Equation is 0, so the predicted nAChRconcentration is Y= 0 = for the schizophrenics in the sample, Y= 1 = reason for calling the General Linear Model General is that it can handleanXthat isnotnumerical as well as one that is numerical. Hence, there isno difference between performing a GLM analysis using Equation withXisvariable Schizophrenia with values of No and Yes and performing one whereXis the numerical variable SzDummyCode with values of 0 and 1.

5 Table the results of GLMs in which theXvariable is the numeric SzDummyCode(top) and in which theXvariable is the qualitative variable that there are no differences in any value between the output for vari-able SzDummyCode and Schizophrenia. Notice also that there the bottom halfof the table labels the variable SchizophreniaYes and not simply Schizophre-nia. This is a hint as to what is going on when the GLM handles a nonnumeric1 Dummy coding is described in Section EXAMPLE WITH MORE THAN ONE PREDICTOR 9. THE General Linear Model (GLM): A GENTLEINTRODUCTIONT able : GLM results using a numeric (SzDummyCode) and a nonnumeric(Schizophrenia) variable SzDummyCodeVariableEstimate St.

6 4E-11 SzDummyCode .496 Nonnumeric variable SchizophreniaVariableEstimate St. 4E-11 SchizophreniaYes .496 Xvariable. All GLM programs change the nonnumeric variable into a nu-meric one so that they can solve the mathematical problem. After that is done,the GLM translates the numerical output back into the original , the SchizophreniaYes using the variable Schizophrenia signifies thatone should add to the value of the intercept to get the predicted valuewhen the variable Schizophrenia = Yes. (A cautionary aside: Different GLM programs use different mechanisms forconverting the categories in a nonnumeric variable into numbers.)

7 Also, a usercan specify how to perform the conversion. Thus, the values of the scanbe different for different coding schemes for the same problem. The predictedvalues, however, for the groups will always remain the same).Finally, look at thepvalue for the effect. It is .496 and definitely non-significant. One might be tempted to conclude that there is no difference innAChR concentrations between schizophrenics and controls, but that would beunwise. To see why, we must combine substantive knowledge on neurosciencewith Example with more than one predictor , schizophrenics smoke a lot. Most of you have already asked yourselfabout the effect of smoking on the nicotinic receptor density.

8 Similarly, smok-ing is associated with early death, so any effect of age on nAChR concentrationmight also cloud the results. These are not trivial issues because there is evi-dence that the number of nicotinic receptors decrease with age (?)andthattheyare upregulated by the use of nicotine (?). The increase in nAChR from smok-ing and early death might have masked the differences between schizophrenicsand controls in this hypothetical 9. THE General Linear Model (GLM): A EXAMPLE WITH MORE THAN ONE PREDICTOR : Results of the GLM predicting nAChR from Age and SzDummyCodeVariableEstimate , one would like to have a control matched to each schizophrenic on ageof death and smoking status at or near death.

9 The practicalities of research withbrain banks, however, make it difficult and expensive perhaps even impossible to pull that off. Smoking status at death is often not known, and even if it isknown, there is wide variability in the amount of nicotine intake among , the data on variable Smoke (was the person a smoker at or near death?)in Table has so many unknowns as to make the variable useless. One way toaddress this issue is to measure brain cotinine, a metabolite of nicotine, becauseit has a longer half-life than now want to control for both age and cotinine levels. We could dividethe specimens into groups by categorizing variables Age and Cotinine, but thatapproach is not recommended.

10 In fact, it is downright stupid. If we used a cutoffof 65 on age for young versus old, there would be no young schizophrenicswith low cotinine values, and we would be comparing groups of size four withthose of size two in other ,however,avoidsthis. Supposethatwewanttocontrolfor Age. We just add a secondXvariable to the right-hand side of , or Y= 0+ 1X1+ 2X2( )It is good practice to put any control variables into the equation before thevariable of interest soX1denotes variable Age andX2is, as before, SzDummy-Code (or Schizophrenia). Instead of a two dimensional plot as in Figure , theproblem would now be visualized via a three dimensional plot.