Research Methods William G. Zikmund Bivariate Analysis - Tests of Differences Common Bivariate Tests Differences among Differences between Type of Measurement three or more two independent groups independent groups Independent groups: One-way Interval and ratio t-test or Z-test ANOVA Common Bivariate Tests Differences among Differences between Type of Measurement three or more two independent groups independent groups Mann-Whitney U-test Ordinal Kruskal-Wallis test Wilcoxon test Common Bivariate Tests Differences among Differences between Type of Measurement three or more two independent groups independent groups Z-test (two proportions) Nominal Chi-square test Chi-square test Type of Differences between Measurement two independent groups Nominal Chi-square test Differences Between Groups • Contingency Tables • Cross-Tabulation • Chi-Square allows testing for significant differences between groups • “Goodness of Fit” Chi-Square Test (Oi − Ei )² x² = Ei x² = chi-square statistics th Oi = observed frequency in the i cell th Ei = expected frequency on the i cell Chi-Square Test R C E i j ij n th Ri = total observed frequency in the i row th Cj = total observed frequency in the j column n = sample size Degrees of Freedom (R-1)(C-1)=(2-1)(2-1)=1 Health Economics Research Method 2003/2 Degrees of Freedom d.f.=(R-1)(C-1) Awareness of Tire Manufacturer’s Brand Men Women Total Aware 50 10 60 Unaware 15 25 40 65 35 100 Chi-Square Test: Differences Among Groups Example (50 − 39 )2 (10 − 21 )2 X 2 = + 39 21 (15 − 26 )2 (25 − 14 )2 + + 26 14 χ 2 = 3.102 + 5.762 + 4.654 + 8.643 = χ 2 = 22 .161 d. f . = (R −1)( C −1) d. f . = 2( − )(1 2 − )1 =1 X2=3.84 with 1 d.f. Type of Differences between Measurement two independent groups Interval and t-test or ratio Z-test Differences Between Groups when Comparing Means • Ratio scaled dependent variables • t-test – When groups are small – When population standard deviation is unknown • z-test – When groups are large Null Hypothesis About Mean Differences Between Groups µ1 − µ 2 OR µ1 − µ 2 = 0 t-Test for Difference of Means mean 1 - mean 2 t = Variabilit y of random means t-Test for Difference of Means Χ − Χ t = 1 2 S X1 − X 2 X1 = mean for Group 1 X2 = mean for Group 2 SX1-X2 = the pooled or combined standard error of difference between means. t-Test for Difference of Means 1 2 X1 − X 2 t-Test for Difference of Means X1 = mean for Group 1 X2 = mean for Group 2 S = the pooled or combined standard error X1-X2 of difference between means. Pooled Estimate of the Standard Error ()n −1S2 +(n − )1 S2) 1 1 S = 1 1 2 2 + X1−X2 n1 +n2 −2 n1 n2 Pooled Estimate of the Standard Error 2 S1 = the variance of Group 1 2 S2 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2 Pooled Estimate of the Standard Error t-test for the Difference of Means ()n −1 S 2 + (n − )1 S 2 ) 1 1 S = 1 1 2 2 + X1 − X 2 n1 + n2 − 2 n1 n2 2 S1 = the variance of Group 1 2 S2 = the variance of Group 2 n1 = the sample size of Group 1 n2 = the sample size of Group 2 Degrees of Freedom • d.f. = n - k • where: –n = n 1 + n2 –k = number of groups t-Test for Difference of Means Example (20 )(1.2 )2 + (13 )(6.2 )2 1 1 S = + X1−X2 33 21 14 = .797 16 5. −12 2. 3.4 t = = .797 .797 = .5 395 Type of Differences between Measurement two independent groups Nominal Z-test (two proportions) Comparing Two Groups when Comparing Proportions • Percentage Comparisons • Sample Proportion - P • Population Proportion - Π Differences Between Two Groups when Comparing Proportions The hypothesis is: Ho: Π1 = Π2 may be restated as: Ho: Π1 − Π2 = 0 Z-Test for Differences of Proportions Ho :π1 = π 2 or Ho :π1 − π 2 = 0 Z-Test for Differences of Proportions p − p − π − π Z = 1 2 1 2 S p1 − p2 Z-Test for Differences of Proportions p1 = sample portion of successes in Group 1 p2 = sample portion of successes in Group 2 (π1 − π1) = hypothesized population proportion 1 minus hypothesized population proportion 1 minus Sp1-p2 = pooled estimate of the standard errors of difference of proportions Z-Test for Differences of Proportions 1 1 S = pq − p1− p2 n1 n2 Z-Test for Differences of Proportions p = pooled estimate of proportion of success in a sample of both groups qp = (1- p) or a pooled estimate of proportion of failures in a sample of both groups n1= sample size for group 1 n2= sample size for group 2 Z-Test for Differences of Proportions n p n p p 1 1 2 2 n1 n2 Z-Test for Differences of Proportions 1 1 S p − p = (.375 )(.625 ) + 1 2 100 100 = .068 A Z-Test for Differences of Proportions (100 )( 35. )+ (100 )( 4. ) p = 100 +100 = .375 Type of Differences between Measurement three or more independent groups Interval or ratio One-way ANOVA Analysis of Variance Hypothesis when comparing three groups µ1 = µ2 = µ3 Analysis of Variance F-Ratio Variance − between − groups F = Variance − within − groups Analysis of Variance Sum of Squares SS total = SS within + SS between Analysis of Variance Sum of SquaresTotal n c 2 SS total = (X ij − X ) i = 1 j =1 Analysis of Variance Sum of Squares th Xpiij = individual scores, i.e., the i observation or test unit in the jth group Xpi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of SquaresWithin n c 2 SS within = (X ij − X j ) i = 1 j =1 Analysis of Variance Sum of SquaresWithin th Xpiij = individual scores, i.e., the i observation or test unit in the jth group Xpi = grand mean n = number of all observations or test units in a group c = number of jth groups (or columns) Analysis of Variance Sum of Squares Between n 2 SS between = n j (X j − X ) j=1 Analysis of Variance Sum of squares Between th X j= individual scores, i.e., the i observation or test unit in the jth group X = grand mean nj = number of all observations or test units in a group Analysis of Variance Mean Squares Between SS MS = between between c −1 Analysis of Variance Mean Square Within SS MS = within within cn − c Analysis of Variance F-Ratio MS F = between MS within A Test Market Experiment on Pricing Sales in Units (thousands) Regular Price Reduced Price Cents-Off Coupon $.99 $.89 Regular Price Test Market A, B, or C 130 145 153 Test Market D, E, or F 118 143 129 Test Market G, H, or I 87 120 96 Test Market J, K, or L 84 131 99 Mean X1=104.75 X2=134.75 X1=119.25 Grand Mean X=119.58 ANOVA Summary Table Source of Variation • Between groups • Sum of squares – SSbetween • Degrees of freedom – c-1 where c=number of groups • Mean squared-MSbetween – SSbetween/c-1 ANOVA Summary Table Source of Variation • Within groups • Sum of squares – SSwithin • Degrees of freedom – cn-c where c=number of groups, n= number of observations in a group • Mean squared-MSwithin – SSwithin/cn-c ANOVA Summary Table Source of Variation • Total • Sum of Squares – SStotal • Degrees of Freedom – cn-1 where c=number of groups, n= number of observations in a group MS BETWEEN F = MS WITHIN Research Methods William G. Zikmund Bivariate Analysis: Measures of Associations Measures of Association • A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables. Relationships Among Variables • Correlation analysis • Bivariate regression analysis Type of Measure of Measurement Association Interval and Correlation Coefficient Ratio Scales Bivariate Regression Type of Measure of Measurement Association Ordinal Scales Chi-square Rank Correlation Type of Measure of Measurement Association Chi-Square Nominal Phi Coefficient Contingency Coefficient Correlation Coefficient • A statistical measure of the covariation or association between two variables. • Are dollar sales associated with advertising dollar expenditures? The Correlation coefficient for two variables, X and Y is xy Health Economics Research Method 2003/2. Correlation Coefficient • r • r ranges from +1 to -1 • r = +1 a perfect positive linear relationship • r = -1 a perfect negative linear relationship • r = 0 indicates no correlation Simple Correlation Coefficient (X − X )(Y − Y ) r = r = ∑ i i xy yx 2 2 ∑()Xi − X ∑ ()Yi − Y Simple Correlation Coefficient σ r = r = xy xy yx 2 2 σ xσ y Simple Correlation Coefficient Alternative Method 2 σ x = Variance of X 2 σ y = Variance of Y σ xy = Covariance of X and Y Y Correlation Patterns NO CORRELATION X Health Economics Research Method 2003/2. Y Correlation Patterns PERFECT NEGATIVE CORRELATION - r= -1.0 X Health Economics Research Method 2003/2. Correlation Patterns Y A HIGH POSITIVE CORRELATION r = +.98 X Health Economics Research Method 2003/2. Calculation of r − .6 3389 r = (17 .837 )(.5 589 ) − .6 3389 = = −.635 99 .712 Pg 629 Coefficient of Determination Explained variance r 2 = Total Variance Correlation Does Not Mean Causation • High correlation • Rooster’s crow and the rising of the sun – Rooster does not cause the sun to rise. • Teachers’ salaries and the consumption of liquor – Covary because they are both influenced by a third variable Correlation Matrix • The standard form for reporting correlational results.