remember you just invented a “magic math pill” that will increase test scores. on the day of the...

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. . .

• 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.

• You have more than 2 groups • You have a repeated measures design

• You need to conduct a Repeated Measures ANOVA

Tests to Compare Means

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

Design of experiment

What if. . .

• 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 74 75 78

For now . . . Ignore that it is a repeated design

Sub 1 57 60 64

Sub 2 71 72 74

Sub 3 75 76 78

Sub 4 93 92 96

Mean 74 75 78

Sub 1 57 60 64

Sub 2 71 72 74

Sub 3 75 76 78

Sub 4 93 92 96

Mean 74 75 78

Between Variability = low

Sub 1 57 60 64

Sub 2 71 72 74

Sub 3 75 76 78

Sub 4 93 92 96

Mean 74 75 78

Within Variability = high

34.667 2 17.333 .091 .914

1720.0 9 191.11

1754.7 11

BetweenGroupsWithinGroupsTotal

Sum ofSquares df

MeanSquare F Sig.

Notice – the within variability of a group can be predicted by the other groups

Sub 1 57 60 64

Sub 2 71 72 74

Sub 3 75 76 78

Sub 4 93 92 96

Mean 74 75 78

Notice – the within variability of a group can be predicted by the other groups

Sub 1 57 60 64

Sub 2 71 72 74

Sub 3 75 76 78

Sub 4 93 92 96

Mean 74 75 78

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

Sub 3 75 76 78 76.33

Sub 4 93 92 96 93.66

Mean 74 75 78

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.

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

Mean 74 75 78 75.66

Example

Sum of Squares

• SS Total

– The total deviation in the observed scores

• Computed the same way as before

2..)( XXSSTotal

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

Mean 74 75 78 75.66

SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 908

*What makes this value get larger?

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

Mean 74 75 78 75.66

SStotal = (57-75.66)2+ (71-75.66)2+ . . . . (96-75.66)2 = 908

*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)

2..)( XXkSS SSubjects

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

Mean 74 75 78 75.66

SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712

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

Mean 74 75 78 75.66

SSSubjects = 3((60.33-75.66)2+ (72.33-75.66)2+ . . . . (93.66-75.66)2) = 1712

*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)

2..)( XXnSS WTreatment

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

Mean 74 75 78 75.66

SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66

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

Mean 74 75 78 75.66

SSTreatment = 4((74-75.66)2+ (75-75.66)2+(78-75.66)2) = 34.66

*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

Compute df

Source df SS

Subjects 1712.00

Treatment 34.66

Error 8.00

Total 11 1754.66

df total = N -1

Compute df

Source df SS

Subjects 3 1712.00

Treatment 34.66

Error 8.00

Total 11 1754.66

df total = N -1

df subjects = n – 1

Compute df

Source df SS

Subjects 3 1712.00

Treatment 2 34.66

Error 8.00

Total 11 1754.66

df total = N -1

df treatment = k-1

Compute df

Source df SS

Subjects 3 1712.00

Treatment 2 34.66

Error 6 8.00

Total 11 1754.66

df total = N -1

df treatment = k-1

df error = (n-1)(k-1)

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

Measure: MEASURE_1Sphericity Assumed

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

SourcePILLSError(PILLS)

Type IIISum ofSquares df

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Tests of Within-Subjects Effects

Computed using alpha = .05a.

Additional tests

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

Can investigate the meaning of the F value by computing t-tests and Fisher’s LSD

Remember

nMSXXtwithin2

Mean 74 75 78 75.66

nMSXXtwithin2

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Mean 74 75 78 75.66

nMSXXtwithin2

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Pill vs. Placebo

Mean 74 75 78 75.66

4)33.1(2

757423.1

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Pill vs. Placebo t=1.23

Mean 74 75 78 75.66

4)33.1(2

757423.1

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

t (6) critical = 2.447

Mean 74 75 78 75.66

4)33.1(2

787498.4

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Pill vs. No Pill t =4.98*

Mean 74 75 78 75.66

4)33.1(2

787570.3

34.667 2 17.333 13.000 .007 26.000 .9408.000 6 1.333

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

Pill vs. No Pill t =4.98*

Placebo vs. No Pill t =3.70*

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. Examine this question and (if there is a difference) determine which tests are significantly different.

Test 1 Test 2 Test 3

Sub 1 60 70 78

Sub 2 78 76 85

Sub 3 64 90 89

Sub 4 77 81 94

564.50 2 282.25 7.211 .025 14.423 .742234.83 6 39.139

SourceFACTOR1Error(FACTOR1)

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

564.50 2 282.25 7.211 .025 14.423 .742234.83 6 39.139

SourceFACTOR1Error(FACTOR1)

MeanSquare F Sig.

Noncent.Parameter

ObservedPowera

4 60.00 78.00 69.750 9.10594 70.00 90.00 79.250 8.46074 78.00 94.00 86.500 6.7577

TEST1TEST2TEST3Valid N(listwise)

N Minimum Maximum MeanStd.

Deviation

Descriptive Statistics

nMSXXtwithin2

SPSS Homework – Bonus

1) Determine if practice had an effect on test scores.2) Examine if there is a linear trend with practice on test scores.

Why is this important?

• Requirement

• Understand research articles

• Do research for yourself

• Real world

The Three Goals of this Course

• 1) Teach a new way of thinking

• 2) Teach “factoids”

2)( ijWithin XXSS

SSCOVr

)(Z . . . . )(Z )(Z Y pp2211Z

)1()1(2

RpRpNF

aar YPS

1e b c a

a 2).(

What you have learned!• Describing and Exploring Data / The

Normal Distribution

• Scales of measurement– Populations vs. Samples

• Learned how to organize scores of one variable using:

– frequency distributions– graphs

What you have learned!

• Measures of central tendency– Mean– Median– Mode

• Variability– Range– IQR– Standard Deviation– Variance

– Z Scores

– Find the percentile of a give score– Find the score for a given percentile

• Sampling Distributions & Hypothesis Testing

– Is this quarter fair?– Sampling distribution

• CLT

– The probability of a given score occurring

What you have learned!• Basic Concepts of Probability

– Joint probabilities– Conditional probabilities

– Different ways events can occur• Permutations• Combinations

– The probability of winning the lottery

– Binomial Distributions• Probability of winning the next 4 out of 10 games of Blingoo

• Categorical Data and Chi-Square

– Chi square as a measure of independence• Phi coefficient

– Chi square as a measure of goodness of fit

• Hypothesis Testing Applied to Means

– One Sample t-tests

– Two Sample t-tests• Equal N• Unequal N• Dependent samples

• Correlation and Regression

– Correlation

– Regression

• Alternative Correlational Techniques

– Pearson Formulas• Point-Biserial• Phi Coefficent• Spearman’s rho

– Non-Pearson Formulas• Kendall’s Tau

• Multiple Regression

– Multiple Regression• Causal Models• Standardized vs. unstandarized • Multiple R• Semipartical correlations

– Common applications• Mediator Models• Moderator Mordels

• Simple Analysis of Variance

– ANOVA

– Computation of ANOVA

– Logic of ANOVA• Variance• Expected Mean Square• Sum of Squares

What you have learned!• Multiple Comparisons Among Treatment Means

– What to do with an omnibus ANOVA• Multiple t-tests• Linear Contrasts• Orthogonal Contrasts• Trend Analysis

– Controlling for Type I errors• Bonferroni t• Fisher Least Significance Difference• Studentized Range Statistic• Dunnett’s Test

• Factorial Analysis of Variance

– Factorial ANOVA

– Computation and logic of Factorial ANOVA

– Interpreting Results• Main Effects• Interactions

What you have learned!• Factorial Analysis of Variance and Repeated

Measures

– Factorial ANOVA

– Computation and logic of Factorial ANOVA

– Interpreting Results• Main Effects• Interactions

– Repeated measures ANOVA

The Three Goals of this Course

• 1) Teach a new way of thinking

• 2) Teach “factoids”

• 3) Self-confidence in statistics

• CRN 33515.0

Four Step When Solving a Problem

• 1) Read the problem

• 2) Decide what statistical test to use

• 3) Perform that procedure

• 4) Write an interpretation of the results

• 1) Read the problem1) Read the problem

• 2) Decide what statistical test to use

• 3) Perform that procedure3) Perform that procedure

• 4) Write an interpretation of the results4) Write an interpretation of the results

• 1) Read the problem

• 2) Decide what statistical test to use2) Decide what statistical test to use

• 3) Perform that procedure

• 4) Write an interpretation of the results

How do you know when to use what?

• If you are given a word problem, would you know which statistic you should use?

Example

• An investigator wants to predict a male adult’s height from his length at birth. He obtains records of both measures from a sample of males.

a. Independent t-test k. Regression b. Dependent t-test l. Standard Deviation c. One-sample t-test m. Z-score d. Goodness of fit Chi-Square n. Mode e. Independence Chi-Square o. Mean f. Dunnett's test p. Median g. Correlation (Pearson r) q. Fisher's LSD h. Scatter Plot r. Binomial Distribution i. Bonferroni t s. ANOVA j. Factorial ANOVA t. Kendall's Tau

Example

• An investigator wants to predict a male adult’s height from his length at birth. He obtains records of both measures from a sample of males.

• Use regression

Type of Data

Qualitative Quantitative

One categorical variable

Two categorical variables

Goodness of FitChi Square

Independence Chi Square

Differences Relationships

One predictor

Two predictors

Continuous measurement

Degree ofRelationship

Prediction

PearsonCorrelation

Regression

Spearmn’s r orKendell’s Tau

Multiple Regression

One Group

Two Groups

Multiple Groups

One samplet-test

IndependentGroups

DependentGroups

Two-samplet-test

Dependentt-test

IndependentGroups

DependentGroups

One IV

Two IVs

One-wayANOVA

FactorialANOVA

Repeated mmeasuresANOVA

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