Remember You just invented a “magic math pill” that will increase test scores. On the day of the first test you give the pill to 4 subjects. When these same subjects take the second test they do not get a pill Did the pill increase their test scores?

What if. . . You just invented a “magic math pill” that will increase test scores. On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

Note You have more than 2 groups You have a repeated measures design You need to conduct a Repeated Measures ANOVA

Tests to Compare Means Design of experiment Independent Variables and # of levels Independent Samples Related Samples One IV, 2 levels Independent t-test Paired-samples t-test One IV, 2 or more levels ANOVA Repeated measures ANOVA Tow IVs, each with 2 or more levels Factorial ANOVA Repeated measures factorial ANOVA

What if. . . You just invented a “magic math pill” that will increase test scores. On the day of the first test you give a full pill to 4 subjects. When these same subjects take the second test they get a placebo. When these same subjects that the third test they get no pill.

Results Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Sub 4 93 92 96 Mean

For now . . . Ignore that it is a repeated design Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Sub 4 93 92 96 Mean

Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 93 92 96 Mean Between Variability = low

Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 93 92 96 Mean Within Variability = high

Source df SS MS F Drug 2 34.67 17.33 .09 Within 9 1720 191.11 Total 11 1754.67

Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Notice – the within variability of a group can be predicted by the other groups Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Sub 4 93 92 96 Mean

Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Notice – the within variability of a group can be predicted by the other groups Pill Placebo No Pill Sub 1 57 60 64 Sub 2 71 72 74 Sub 3 75 76 78 Sub 4 93 92 96 Mean Pill and Placebo r = .99; Pill and No Pill r = .99; Placebo and No Pill r = .99

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 These scores are correlated because, in general, some subjects tend to do very well and others tended to do very poorly

Repeated ANOVA Some of the variability of the scores within a group occurs due to the mean differences between subjects. Want to calculate and then discard the variability that comes from the differences between the subjects.

Example Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 Sub 3 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66

Sum of Squares SS Total Computed the same way as before The total deviation in the observed scores Computed the same way as before

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 1754.67 *What makes this value get larger?

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 1754.67 *What makes this value get larger? *The variability of the scores!

Sum of Squares SS Subjects Represents the SS deviations of the subject means around the grand mean Its multiplied by k to give an estimate of the population variance (Central limit theorem)

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712 *What makes this value get larger?

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712 *What makes this value get larger? *Differences between subjects

Sum of Squares SS Treatment Represents the SS deviations of the treatment means around the grand mean Its multiplied by n to give an estimate of the population variance (Central limit theorem)

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66 *What makes this value get larger?

Pill Placebo No Pill Mean Sub 1 57 60 64 60.33 Sub 2 71 72 74 72.33 75 76 78 76.33 Sub 4 93 92 96 93.66 75.66 SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66 *What makes this value get larger? *Differences between treatment groups

Sum of Squares Have a measure of how much all scores differ SSTotal Have a measure of how much this difference is due to subjects SSSubjects Have a measure of how much this difference is due to the treatment condition SSTreatment To compute error simply subtract!

Sum of Squares SSError = SSTotal - SSSubjects – SSTreatment 8.0 = 1754.66 - 1712.00 - 34.66

Source df SS Subjects 1712.00 Treatment 34.66 Error 8.00 Total 11 1754.66

Source df SS MS F Drug 2 34.67 17.33 .09 Within 9 1720 191.11 Total 11 1754.67

Compute df Source df SS Subjects 1712.00 Treatment 34.66 Error 8.00 df total = N -1 Source df SS Subjects 1712.00 Treatment 34.66 Error 8.00 Total 11 1754.66

Compute df Source df SS Subjects 3 1712.00 Treatment 34.66 Error 8.00 df total = N -1 df subjects = n – 1 Source df SS Subjects 3 1712.00 Treatment 34.66 Error 8.00 Total 11 1754.66

Compute df Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error df total = N -1 df subjects = n – 1 df treatment = k-1 Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error 8.00 Total 11 1754.66

Compute df Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error 6 df total = N -1 df subjects = n – 1 df treatment = k-1 df error = (n-1)(k-1) Source df SS Subjects 3 1712.00 Treatment 2 34.66 Error 6 8.00 Total 11 1754.66

Compute MS Source df SS MS Subjects 3 1712.00 Treatment 2 34.66 17.33 Error 6 8.00 Total 11 1754.66

Compute MS Source df SS MS Subjects 3 1712.00 Treatment 2 34.66 17.33 Error 6 8.00 1.33 Total 11 1754.66

Compute F Source df SS MS F Subjects 3 1712.00 Treatment 2 34.66 17.33 13.00 Error 6 8.00 1.33 Total 11 1754.66

Test F for Significance Source df SS MS F Subjects 3 1712.00 Treatment 2 34.66 17.33 13.00 Error 6 8.00 1.33 Total 11 1754.66

Test F for Significance Source df SS MS F Subjects 3 1712.00 Treatment 2 34.66 17.33 13.00* Error 6 8.00 1.33 Total 11 1754.66 F(2,6) critical = 5.14

Practice You wonder if the statistic tests are of all equal difficulty. To investigate this you examine the scores 4 students got on the three different tests

Test 1 Test 2 Test 3 Sub 1 60 70 78 Sub 2 76 85 Sub 3 64 90 89 Sub 4 77 81 94

Source df SS MS F Subjects Treatment 564.50 Error 234.83 Total 1165.50

Source df SS MS F Subjects 3 366.17 Treatment 2 564.50 282.25 7.21* Error 6 234.83 39.13 Total 11 1165.50

Practice Sleep researchers decide to test the impact of REM sleep deprivation on a computerized assembly line task. Subjects are required to participate in two nights of testing. On each night of testing the subject is allowed a total of four hours of sleep. However, on one of the nights, the subject is awakened immediately upon achieving REM sleep. Subjects then took a cognitive test which assessed errors in judgment. Did sleep deprivation effect subjects cognitive ability?

REM Deprived Control Condition 26 20 15 4 8 9 44 36

tobs = 3.04* Sleep deprivation effected their cognitive abilities.

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