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 But here is the formula == so what you did was 70 + 80+ 80+ 90 = 320 320 / 4 = 80

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 Variability 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

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 Common applications 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

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

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 = 908 *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 = 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)

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

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

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

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

Pill Placebo No Pill Mean 74 75 78 75.66

Pill Placebo No Pill Mean 74 75 78 75.66 Pill vs. Placebo

Pill Placebo No Pill Mean 74 75 78 75.66 Pill vs. Placebo t=1.23

Pill Placebo No Pill Mean 74 75 78 75.66 Pill vs. Placebo t=1.23 t (6) critical = 2.447

Pill Placebo No Pill Mean 74 75 78 75.66 Pill vs. Placebo t=1.23 Pill vs. No Pill t =4.98* t (6) critical = 2.447

Pill Placebo No Pill Mean 74 75 78 75.66 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 1 Test 2 Test 3 Sub 1 60 70 78 Sub 2 76 85 Sub 3 64 90 89 Sub 4 77 81 94

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.

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 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 problem 2) 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.

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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