An Introduction to Classification• Classification vs. Prediction• Classification & ANOVA• Classification Cutoffs, Errors, etc.• Multivariate Classification & Linear Discriminant FunctionAn Introduction to ClassificationLet’s start by reviewing what “prediction” is…Using a person’s scores on one or more variables to make a “best guess” of the that person’s score on another variable (the value of which isn’t known)Classification is very similar …Using a person’s scores on one or more variables to make a “best guess” of the category to which that person belongs (when the category type isn’t known).The difference -- a language “convention”• if the “unknown variable” is quantitative -- its called prediction• if the “unknown variable” is qualitative -- its called classificationHow does classification work???Let’s start with an “old friend” -- ANOVAIn its usual form…• There are two qualitatively different IV groups • naturally occurring or “created” by manipulation• A quantitative DV• H0: MeanG1= Mean G2• Rejecting H0: tells us• There is a relationship between the grouping and DV• Groups represent populations with different means on the DV• Knowing what group a person in allows us to guess their DV score -- mean of that groupLet’s review in a little more detail…Remember the formula for the ANOVA F-testvariation between groups size of the mean differenceF = ----------------------------------- = ---------------------------------------variation within groups variation within groupsIn words -- F compares the mean difference to the variability around each of those meansWhich of the following will produce the larger F-test ? The two data sets have the same means, mean difference & N but the difference is….Data #1 (@ n = 50)group 1 mean = 30std dev = 5group 2 mean = 50std dev = 5Data #2 (@ n = 50)group 1 mean = 30std dev = 15group 2 mean = 50std dev = 15Graphical depictions of these data show that the size of F relates tothe amount of overlap between the groups0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 80 Notice: Since all the distributions have n=50, those with more variabilityare not as tall -- all 4 distributions have the same areaData #1Data #2Larger F = more consistent grp difSmaller F = less consistent grp difLet’s consider that last one “in reverse”…Could knowing the person’s score help tell us what qualitative group they are in? …to “classify” them to the proper group?an Example…Research has revealed a statistical relationship between the number of times a person laughs out loud each day (quant variable) and whether they are depressed or schizophrenic (qual grouping variable).Mean laughsDepressed= 4.0 Mean laughsSchizophrenic= 7.0F(1,34) = 7.00, p < .05A new (as yet undiagnosed) patient laughs 11 times the first daywhat’s your “classification” depressed or schizophrenic?Another patient laughs 1 time -- your “classification”?A third new patient laughs 5 times -- your “classification”?Why were the first two “gimmies” and the last one not?• When the groups have a mean difference, a score beyond one of the group means is more likely to belong to that groupthan to belong to the other group (unless stds are huge)• someone who laughs more than the mean for the schizophrenic group is more likely to be schizohrenicthan to be depressed • someone who laughs less than the mean of the depressive group is more likely to be depressedthan to be schizophrenic• Even when the groups have a mean difference, a score between the group means is harder to correctly classify (unless stdsare miniscule)• someone with 5-6 laughs are hardest to classify, because several depressed and schizophrenic folks have this scoreHere’s a graphical depiction of the clinical data...X 18 dep. patients x x x mean laughs = 4.0 x x x x xx x x x x x x x xlaughs --> 0 1 2 3 4 5 6 7 8 9 0 1 2oo o o 18 schiz. patientso o o o o mean laughs = 7.0o o o o o o o o oLooking at this, its easy to see why we would be ...• confidant in an assignment based on 11 laughs• no depressed patients had a score that high• confident in an assignment based on 1 laugh• no schizophrenic patients had a score that low• lacking confidence in an assignment based on 5 or 6 laughs• several depressed & schizophrenic patients had 5 or 6The process of prediction required two things…• that there be a linear relationship between the predictor and the criterion (reject H0: r = 0)• a formula (y’ = bx + a) to “translate” a predictor score into an estimate of a criterion variable scoreSimilarly, the process of classification requires two things …• a statistical relationship between the predictor (DV) & criterion (reject H0: M1= M2)• a cutoff to “translate” a person’s score on the predictor (DV) into an assignment to one group or the other• where should be place the cutoff??? • Wherever gives us the most accurate classification !!X 18 dep. patients x x x mean laughs = 4.0 x x x x xx x x x x x x x xlaughs --> 0 1 2 3 4 5 6 7 8 9 0 1 21 1 1oo o o 18 schiz. patientso o o o o mean laughs = 7.0o o o o o o o o oWhen your groups are the same size and your group score distributions are symmetrical, things are pretty easy…• place the cutoff at a position equidistant from the group means• here, the cutoff would be 5.5 -- equidistant between 4.0 and 7.0• anyone who laughs more than 5.5 times would be “assigned”as schizophrenic• anyone who laughs fewer than 5.5 times would be “assigned”as depressedx 18 dep. patients x x x mean laughs = 4.0 x x x x xx x x x x x x x xlaughs --> 0 1 2 3 4 5 6 7 8 9 0 1 21 1 1oo o o 18 schiz. patientso o o o o mean laughs = 7.0o o o o o o o o oWe can assess the accuracy of the assignments by building a “reclassification table”Actual DiagnosisAssignment Depressed SchizophrenicDepressed 14 4Schizophrenic 4 14reclassification accuracy would be 28/36 = 77.78%When considering simple regression/prediction, we