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 Independent Variables and # of levels Independent SamplesRelated Samples One IV, 2 levelsIndependent t-testPaired-samples t-test One IV, 2 or more levelsANOVARepeated measures ANOVA Tow IVs, each with 2 or more levels Factorial ANOVARepeated measures factorial ANOVA Design of experiment

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 PillPlaceboNo Pill Sub Sub Sub Sub Mean747578

For now... Ignore that it is a repeated design PillPlaceboNo Pill Sub Sub Sub Sub Mean747578

PillPlaceboNo Pill Sub Sub Sub Sub Mean Between Variability = low

PillPlaceboNo Pill Sub Sub Sub Sub Mean Within Variability = high

Notice – the within variability of a group can be predicted by the other groups PillPlaceboNo Pill Sub Sub Sub Sub Mean747578

Notice – the within variability of a group can be predicted by the other groups PillPlaceboNo Pill Sub Sub Sub Sub Mean Pill and Placebo r =.99; Pill and No Pill r =.99; Placebo and No Pill r =.99

PillPlaceboNo PillMean Sub Sub Sub Sub Mean 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.

PillPlaceboNo PillMean Sub Sub Sub Sub Mean Example

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

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS total = ( ) 2 + ( ) ( ) 2 = 908 *What makes this value get larger?

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS total = ( ) 2 + ( ) ( ) 2 = 908 *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)

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS Subjects = 3(( ) 2 + ( ) ( ) 2) = 1712 *What makes this value get larger?

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS Subjects = 3(( ) 2 + ( ) ( ) 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)

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS Treatment = 4(( ) 2 + ( ) 2 +( ) 2) = *What makes this value get larger?

PillPlaceboNo PillMean Sub Sub Sub Sub Mean SS Treatment = 4(( ) 2 + ( ) 2 +( ) 2) = *What makes this value get larger? *Differences between treatment groups

Sum of Squares Have a measure of how much all scores differ –SS Total Have a measure of how much this difference is due to subjects –SS Subjects Have a measure of how much this difference is due to the treatment condition –SS Treatment To compute error simply subtract!

Sum of Squares SS Error = SS Total - SS Subjects – SS Treatment 8.0 =

Compute df SourcedfSS Subjects Treatment34.66 Error8.00 Total df total = N -1

Compute df SourcedfSS Subjects Treatment34.66 Error8.00 Total df total = N -1 df subjects = n – 1

Compute df SourcedfSS Subjects Treatment Error8.00 Total df total = N -1 df subjects = n – 1 df treatment = k-1

Compute df SourcedfSS Subjects Treatment Error68.00 Total df total = N -1 df subjects = n – 1 df treatment = k-1 df error = (n-1)(k-1)

Compute MS SourcedfSSMS Subjects Treatment Error68.00 Total

Compute MS SourcedfSSMS Subjects Treatment Error Total

Compute F SourcedfSSMSF Subjects Treatment Error Total

Test F for Significance SourcedfSSMSF Subjects Treatment Error Total

Test F for Significance SourcedfSSMSF Subjects Treatment * Error Total F(2,6) critical = 5.14

Additional tests SourcedfSSMSF Subjects Treatment * Error Total Can investigate the meaning of the F value by computing t-tests and Fisher’s LSD

Remember

PillPlaceboNo PillMean

PillPlaceboNo PillMean Pill vs. Placebo

PillPlaceboNo PillMean Pill vs. Placebo t=1.23

PillPlaceboNo PillMean Pill vs. Placebo t=1.23 t (6) critical = 2.447

PillPlaceboNo PillMean Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* t (6) critical = 2.447

PillPlaceboNo PillMean Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* Placebo vs. No Pill t =3.70* t (6) critical = 2.447

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 1Test 2Test 3 Sub Sub Sub Sub

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”

Mean

r =

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

What you have learned! –Z Scores –Find the percentile of a give score –Find the score for a given percentile

What you have learned! 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

What you have learned! Categorical Data and Chi-Square –Chi square as a measure of independence Phi coefficient –Chi square as a measure of goodness of fit

What you have learned! Hypothesis Testing Applied to Means –One Sample t-tests –Two Sample t-tests Equal N Unequal N Dependent samples

What you have learned! Correlation and Regression –Correlation –Regression

What you have learned! Alternative Correlational Techniques –Pearson Formulas Point-Biserial Phi Coefficent Spearman’s rho –Non-Pearson Formulas Kendall’s Tau

What you have learned! Multiple Regression –Multiple Regression Causal Models Standardized vs. unstandarized Multiple R Semipartical correlations –Common applications Mediator Models Moderator Mordels

What you have learned! 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

What you have learned! 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

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

Four Step When Solving a Problem 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

Four Step When Solving a Problem 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.

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 One categorical variable Goodness of Fit Chi Square Two categorical variables Independence Chi Square Quantitative Differences One Group One sample t-test Two Groups Independent Groups Two-sample t-test Dependent Groups Dependent t-test Multiple Groups Independent Groups One IV One-way ANOVA Two IVs Factorial ANOVA Dependent Groups Repeated mmeasures ANOVA Relationships One predictor Continuous measurement Degree of Relationship Pearson Correlation Prediction Regression Ranks Spearmn’s r or Kendell’s Tau Two predictors Multiple Regression