Stat 401 B – Lecture 331Categorical Variables Response: Highway MPG Explanatory: Type of drive All Wheel Rear Wheel Front Wheel2Indicator Variables We have used indicator variables so that we can trick JMP into analyzing the data using multiple regression.3Categorical Variables There is a more straight forward analysis that can be done with categorical explanatory variables.Stat 401 B – Lecture 334Categorical Variables The analysis is an extension of the two independent sample analysis we did at the beginning of the semester. Body mass index for men and women (Lectures 4 and 5).5Analysis of Variance Response: numerical, Y Explanatory: categorical, X Total Sum of Squares()∑−=2yySSTotal6Sum of Squares Total()990.36697.272==−==∑dfyySSyTotalStat 401 B – Lecture 337Analysis of Variance Partition the Total Sum of Squares into two parts. Due to differences among the sample means for the categories. Due to variation within categories, i.e error variation.8Sum of Squares Factor()iyinyynSSiiiiFactorcategory for mean samplecategory in nsobservatio ofnumber 2==−=∑9Category Sample Means6029.983Front Wheel1726.529Rear Wheel2322.608All WheelSample SizeMeanStat 401 B – Lecture 3310Sum of Squares Drive()()()()3.9327.27983.2960 7.27529.2617 7.27608.22232222=−+−+−=−=∑DriveDriveiiFactorSSSSyynSS11Sum of Squares Error()isinsnSSiiiiErrorcategory for variancesamplecategory in nsobservatio ofnumber 122==−=∑12Category Sample Variances6037.881Front Wheel179.890Rear Wheel2315.613All WheelSample SizeVarianceStat 401 B – Lecture 3313Sum of Squares Error()()()()7.2736881.3759 89.916 613.152212=++=−=∑ErrorErroriiErrorSSSSsnSS14Mean Square A mean square is the sum of squares divided by its associated degrees of freedom. A mean square is an estimate of variability.15Mean Square Factor The mean square factor estimates the variability due to differences among category sample means. If the mean square factor is large, this indicates the category sample means are quite different.Stat 401 B – Lecture 3316Mean Square Error The mean square error estimates the naturally occurring variability, i.e. the error variance, . This is the ruler against which the variability among sample means is measured.2σ17Test of Hypothesis H0: all the category population means are equal. HA: some of the category population means are not equal. Similar to the test of model utility.18Test Statistic F = MSFactor/ MSError P-value = Prob > F If the P-value is small, reject H0and declare that at least two of the categories have different population means.Stat 401 B – Lecture 3319Analysis of VarianceSSTotalN – 1TotalMSErrorSSErrorN – kErrorMSFactor/ MSErrorMSFactorSSFactork – 1Factor (Model)FMSSSdfSource20Analysis of Variance3669.099Total28.2132736.797Error16.52466.15932.32Factor (Model)FMSSSdfSource21Test of Hypothesis F = 16.52, P-value < 0.0001 Because the P-value is so small, there are some categories that have different population means.Stat 401 B – Lecture 3322JMP Response, Y: Highway MPG (numerical) Explanatory, X: Drive (categorical) Fit Y by X231015202530354045505560Highway MPGAll Front RearDriveOneway Analysis of Highway MPG By Drive24JMP Fit Y by X From the red triangle pull down menu next to Oneway select Means/Anova Display options  Uncheck Mean Diamonds Check Mean LinesStat 401 B – Lecture 3325Test of Significance The test of significance, like the test of model utility, is very general. We know there are some categories with different population means but which categories are they?26Multiple Comparisons In ANOVA, a statistically significant F test is often followed up by a procedure for comparing the sample means of the categories.27Least Significant Difference One multiple comparison method is called Fisher’s Least Significant Difference, LSD. This is the smallest difference in sample means that would be declared statistically significant.Stat 401 B – Lecture 3328Least Significant DifferencejinndftnnRMSEtLSDjiErrorji and categoriesfor sizes sample theare and freedom of degrees andconfidence 95%for valuea is where11**=+=29Least Significant Difference When the number of observations in each category is the same, there is one value of LSD for all comparisons. When the numbers of observations in each category are different, there is one value of LSD for each comparison.30Compare All to Rear Wheel All Wheel: ni= 23 Rear Wheel: nj= 17 t*= 1.9847 RMSE = 5.3116Stat 401 B – Lecture 3331Compare All to Rear Wheel()372.31712313116.59847.1=⎟⎠⎞⎜⎝⎛+=LSDLSD32Compare All to Rear Wheel All Wheel: mean = 22.609 Rear Wheel: mean = 26.529 Difference in means = 3.92 3.92 is bigger than the LSD = 3.37, therefore the difference between All Wheel and Rear Wheel is statistically significant.33JMP – Fit Y by X From the red triangle pull down menu next to Oneway select Compare Means – Each Pair Student’s t.Stat 401 B – Lecture 33341.98472t0.05AlphaFrontRearAll-1.92470.55744.78920.5574-3.61590.54894.78920.5489-3.1087Abs(Dif)-LSDFront Rear AllPositive values show pairs of means that are significantly different.FrontRearAllLevelA B C29.98333326.52941222.608696MeanLevels not connected by same letter are significantly different.FrontRearFrontLevelAllAllRear- Level7.3746383.9207163.453922Difference4.7892430.5488600.557427Lower CL9.9600337.2925726.350416Upper CL<.0001*0.0231*0.0199*p-ValueComparisons for each pair using Student's tMeans ComparisonsOneway Analysis of Highway MPG By Drive35Regression vs ANOVA Note that the P-values for the comparisons are the same as the P-values for the slope estimates in the regression on indicator variables.36Regression vs ANOVA Multiple regression with indicator variables and ANOVA give you exactly the same