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2nd Half Review ANOVA (Ch. 11) Non-Parametric (7.11, 9.5) Regression (Ch. 12) ANCOVA Categorical (Ch. 10) Correlation (Ch. 12) The Exam • Thursday, April 27, 9:00am – TC 348 Abdi to Middleton – TC 348a Minto to Shetty – TC 348b Siddiqui to Zdravic • • • • • 50 Questions Not Cumulative 3hr Bring a calculator No formula Sheets ANOVA • Variance Partitions – Total = Among + Within • Grand Mean, Group Mean and associated Deviations • When do we reject based on variance ratio??? ANOVA Table Source df Among Treatments k-1 SS n X k j j 1 Within Treatments n-k k j X MS 2 n 2 ( X X ) ij j j 1 i 1 Total n-1 k n 2 ( X X ) ij j 1 i 1 F SSamong MSamong dfamong MSwithin SSwithin dfwithin ANOVA • • • • When do we use?? Model I vs Model II vs Model III?? Multi-Factors?? Main Effects vs Interactions?? Example From Text Question #11.40, p. 518 10 women in an aerobic exercise class, 10 women in a modern dance class, and a control group of 9 women were studied. One measurement made on each women was change in fat-free mass over the course of the 16-week training period. Summary statistics are given in the table. The ANOVA SS(between) is 2.465 and the SS(within) is 50.133. Aerobics Dance Control Mean 0 0.44 0.71 SD 1.31 1.17 1.68 n 10 10 9 a) State the null hypothesis b) Construct the ANOVA table and test the null hypothesis (α = 0.05) Example From Text Question #11.57, p. 522 A new investigational drug was given to 4 male and 4 female dogs, at doses 8 mg/kg and 25 mg/kg. The variable recorded was alkaline phosphatase level (U/Li). Dose (mg/kg) Male Female 8 171 150 154 127 104 152 143 105 143 133.5 80 101 149 113 138 161 131 197 124.5 143 Avg 25 Avg (SS(sex) = 81, SS(dose) = 81, SS(interaction) = 784, and SS(within) = 12604 a) Construct the ANOVA Table b) Carry out an F test for interactions: use (α = 0.05). c) Test the null hypothesis that does has no effect on alkaline phosphatase level. (α = 0.05) Extra Questions from the Text • • • • 11.4-11.6 11.9-11.11 11.17, 11.19 11.42, 11.43, 11.50, 11.54 Non-Parametric • When to use?? – Normality – Homogenous of Variance – Independent Observations • What about Power?? • What do they use to compare data?? Mann-Whitney Test • Compares two samples • Replaces two-sample t-test n1 ( n1 1) U n1n2 R1 2 n2 ( n2 1) U n2 n1 R2 2 If either U or U’ is greater than the critical value of U, then you should reject the Ho Critical U UCritical U0.05,( 2 ),n1 ,n2 UCritical U0.05,( 2 ),n2 ,n1 if n1 < n2 if n1 > n2 If either U or U’ is greater than the critical value of U, then you should reject the Ho One-tailed Mann-Whitney U test Use U or U’ depending on whether you expect sample 1 or sample 2 to be bigger Ho: G1 G2 HA: G1 < G2 Ho: G1 G2 HA: G1 > G2 Ranking is low to high U U’ Ranking is high to low U’ U Wilcoxon paired sample test • Compares two paired samples • Replaces Paired t-test Deer 1 2 3 4 5 6 7 8 9 10 Front Leg 142 140 144 144 142 146 149 150 142 148 Back Leg 138 136 147 139 143 141 143 145 136 146 Diff 4 4 -3 5 -1 5 6 5 6 2 Rank |d| Signed Rank |d| Critical T Sum the positive ranks - T+ Sum the negative ranks - TIf either T+ or T- is less than or equal to T0.05, (2),n then reject Ho Can also do these one tailed: Ho: Measurement 1 Measurement 2 HA: Measurement 1 > Measurement 2 --> reject Ho if T- T0.05, (1),n Ho: Measurement 1 Measurement 2 HA: Measurement 1 < Measurement 2 --> reject Ho if T+ T0.05, (1),n Kruskal Wallis Test • Test for three or more groups • Replaces ANOVA k 2 i 12 R H 3( N 1) N ( N 1) i1 ni • Rank each individual sample across all groups • Sum ranks within each group = R • Critical value - 02.05,k 1 Correction factor for tied ranks C 1 t N N 18 1 3 24 24 1 0.0013 0.9987 3 m 3 t t i ti i 1 23 2 23 2 23 2 18 H 44123 . Hc 44180 . C 0.9987 Critical Value Practice Questions • Mann-Whitney – 7.79, 7.80, 7.82-7.84 • Wilcoxon – 9.30 – 9.33 • Kruskal-Wallis – 11.54, 11.57 use Kruskal-Wallis instead of ANOVA Regression • Two or more continuous variables • Linear relationship between Y 0 1 X Intercept Slope Least Squares • Line with smallest residual sum of squares Residual is to ˆ Yi Yˆ as is to X ANOVA Table Source of Variation Regression Residual Total SS DF 2 ( Y Y ) i 2 (Yi Y ) (Y Y ) 2 i 1 n-2 n-1 MS F SSRe gression MSRe gression DFRe gression MSRe sidual SS Re sidual DFRe sidual Coefficient of Determination r 2 SS Re gression SSTotal - proportion of variation explained i a c d e d f i t c B e i M t E g 1 ( 7 3 4 1 D 2 0 8 3 0 a D Intercept Slope Y 1047 . 0.058 X Confidence Interval for Slope ˆ ˆ ˆ Y 0 1 X 95%CI for 1 ˆ1 t ( 2 ),n 2 * sˆ Practice Questions • 12.45 • 12.49-12.54 ANCOVA • Continuous Dependent • Continuous and Discrete Independents • Compares relationship of two variables across two groups Categorical Data • Discrete Response variable • Interested in Frequencies Chi-Squared Test • Observed vs Expected Frequencies k 2 ( f observed f exp ected ) i 1 2 f exp ected f expected is hypothesized ratio e.g. 50:50 males to females 2 critical 2 0.05, k 1 Contingency Table • Test for independence among variables • 2x2, 2x3, 3x3 etc. • 2 variables with 2 levels, 2 variables with 3 levels, 3 variables with 3 levels etc. k 2 ( f observed f exp ected ) i 1 2 critical f exp ected 2 0.05, k 1 2 Column 1 Variable A Column 2 Column 3 Total Row 1 O11 O21 O31 R1 Row 2 O12 O22 O31 R2 C1 C2 C3 Total Variable B R1 = sum of observed in Row 1 R2 = sum of observed in Row 2 C1 = sum of observed in Column 1 C2 = sum of observed in Column 2 C3 = sum of observed in Column 3 Total = sum of all observed Column 1 Variable A Column 2 Column 3 Total Row 1 E11 E21 E31 R1 Row 2 E12 E22 E32 R2 C1 C2 C3 Total Variable B Expected calculation Eij Ri * C j */ Total Practice Questions • 10.73-10.78 • 10.84-10.88 Logistic Regression • Discrete dependent – usually dichotomous • Continuous independent