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Outline

Outline. Comparing Group Means Data arrangement Linear Models and Factor Analysis of Variance (ANOVA) Partitioning Variance F-test (Computation) T-test and ANOVA Conclusion. Comparing group means. Compares the means of independent variables x1 and x2 (independent sample t-test)

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Outline

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  1. Outline • Comparing Group Means • Data arrangement • Linear Models and Factor • Analysis of Variance (ANOVA) • Partitioning Variance • F-test (Computation) • T-test and ANOVA • Conclusion (c) 2007 IUPUI SPEA K300 (4392)

  2. Comparing group means • Compares the means of independent variables x1 and x2 (independent sample t-test) • Compares two group means of a variable x; examines how group makes difference in x. • Data arrangements: SPSS requires the second arrangement for independent sample t-test (c) 2007 IUPUI SPEA K300 (4392)

  3. Data Arrangement (c) 2007 IUPUI SPEA K300 (4392)

  4. Linear Model and Factor • Examines how group makes difference in the variable x. • X = µ + gi + ei • µ is the overall mean, gi is group i’s mean difference from the overall mean • G is called factor (categorical independent variable) that makes difference in the left-hand side variable (dependent variable) x. (c) 2007 IUPUI SPEA K300 (4392)

  5. Comparing More Variables • What if you need to compare x1, x2, and x3? Or compares means of three groups? • Compares x1 and x2, x1 and x3, and x2 and x3? • How about comparing 4 and 5 groups? • Any way to make comparison easy? • ANOVA is the answer (c) 2007 IUPUI SPEA K300 (4392)

  6. T-test and ANOVA 1 • T-test directly compares means of two variables (groups) • ANOVA (Analysis of Variance) partitions overall variance and examines the impact of factors on mean difference • So, t-test is a special case of ANOVA that considers only TWO GROUPS (c) 2007 IUPUI SPEA K300 (4392)

  7. Data Arrangement: xij • An observation of i group and jth data point xij (c) 2007 IUPUI SPEA K300 (4392)

  8. Partitioning of Variance 1 • Overall mean x** ; group i mean xi* • Sum of squares between group (treatment) • Sum of squares within group • SST = SSM (between) + SSE (within) (c) 2007 IUPUI SPEA K300 (4392)

  9. Partitioning of Variance 2: µ=7.7 (c) 2007 IUPUI SPEA K300 (4392)

  10. ANOVA: F test 1 • H0: all group have the same mean • Ratio of MSM to MSE • Degrees of freedom 1 is t-1 (t is the number of groups) • Degrees of freedom 2 is N-t (N is the number of overall observation) (c) 2007 IUPUI SPEA K300 (4392)

  11. ANOVA: F test 2 • T=3 (three groups); N=15 (5 in each group) • SST=264.9=160.1+104.8 • SSM=160.1, MSM=80.1=160/(3-1) • SSE=104.8, MSE=8.7=104.8/(15-3) • F=9.2=80.1/8.7 (CV=5.10 p.639) • df1=2=3-1, df2=12=15-3 (c) 2007 IUPUI SPEA K300 (4392)

  12. ANOVA: F-test 3 • If F score is larger than the critical value or the p-value is smaller than the significance level, reject the null hypothesis • Rejection of H0 is interpreted as there is at least one group that has a mean different from other group means. • Rejection of H0 does not, however, say which groups have different means. (c) 2007 IUPUI SPEA K300 (4392)

  13. ANOVA: Short Cuts • Compute sum of observations (overall and individual groups) or overall mean x** • Compute variances of groups (si)2 (c) 2007 IUPUI SPEA K300 (4392)

  14. Independent sample t-test • X1bar=$26,800, s1=$600, n1=10 • X2bar=$25,400, s2=$450, n2=8 • Since 5.47>2.58 and p-value <.01, reject the H0 at the .01 level. (c) 2007 IUPUI SPEA K300 (4392)

  15. ANOVA for two group means • sum1=268,000=26,800*10, s1=$600, n1=10 • sum2=203,200=24,400*8, s2=$450, n2=8 • Overall sum=471,200=268,000+203,200, N=18 • SST=SSM+SSE=13,368,611=8,711,111+4,657,500 • F=MSM/MSE=29.9254; sqrt(29.9254)=5.4704=t (c) 2007 IUPUI SPEA K300 (4392)

  16. T-test versus ANOVA • T-test examines mean difference using t distribution (mean difference/standard deviation) • ANOVA examines the mean difference by partitioning variance components of between and within group (F-test) • T-test is a special case of ANOVA • T score is the square root of F score when df1 is 1 (comparing two groups) (c) 2007 IUPUI SPEA K300 (4392)

  17. Summary of Comparing Means (c) 2007 IUPUI SPEA K300 (4392)

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