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Review of Terminology

Review of Terminology. Statistics Parameters Critical Region “Obtained” test statistic “Critical” test statistic Alpha/Confidence Level. Significance Testing. Old way

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Review of Terminology

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  1. Review of Terminology • Statistics • Parameters • Critical Region • “Obtained” test statistic • “Critical” test statistic • Alpha/Confidence Level

  2. Significance Testing • Old way • Find “critical value” of test statistic The point at which the odds of finding that statistic under the null are less than alpha. • Compare your obtained test statistic with the critical test statistic. If your obtained is greater than your critical, reject the null. • Odds of finding that “obtained” value are less than alpha (5%, 1%) if the null is true. • SPSS • Look at “sig” value (aka, “p” value) • Assuming the null is true, there is an X percent chance of obtaining this test statistic. • If it is less than alpha, reject null

  3. T-Tests • 1 sample t-test (univariate t-test) • Compare sample mean of a single I/R variable to a known population mean • Assumes knowledge of population mean (rare) • 2-sample t-test (bivariate t-test) • Compare two sample means (very common) • Dummy IV and I/R Dependent Variable • Difference between means across categories of IV • Do males and females differ on #hours watching TV?

  4. The t distribution • Unlike Z, the t distribution changes with sample size (technically, df) • As sample size increases, the t-distribution becomes more and more “normal” • At df = 120, tcritical values are almost exactly the same as zcritical values

  5. t as a “test statistic” • All test statistics indicate how different our finding is from what is expected under null • Mean differences under null hypothesis = 0 • t indicates how different our finding is from zero • There is an exact “sig” or “p” value associated with every value of t • SPSS generates the exact probability associated with your obtained t

  6. t-score is “meaningful” • Measure of difference in numerator (top half) of equation • Denominator = convert/standardize difference to “standard errors” rather than original metric • Imagine mean differences in “yearly income” versus differences in “# cars owned in lifetime” • Very different metric, so cannot directly compare (e.g., a difference of “2” would have very different meaning) • t = the number of standard errors that separates means • One sample = x versus µ • Two sample = xmales vs. xfemales

  7. t-testing in SPSS • Analyze compare means  independent samples t-test • Must define categories of IV (the dummy variable) • How were the categories numerically coded? • Output • Group Statistics = mean values • Levine’s test • Not real important, if significant, use t-value and sig value from “equal variances not assumed” row • t = “tobtained” • no need to find “t-critical” as SPSS gives you “sig” or the exact probability of obtaining the tobtained under the null

  8. SPSS t-test example • Independent Samples t Test Output: • Testing the Ho that there is no difference in GPA between white and nonwhite UMD students • Is there a difference in the sample?

  9. Interpreting SPSS Output • Difference in GPA across Race? • Obtained t value? Degrees of freedom? • Obtained p value? Specific meaning of p-value? • Reject null?

  10. SPSS and 1-tail / 2-tail • SPSS only reports “2-tailed” significant tests • To obtain a 1-tail test simple divide the “sig value” in half • Sig. (2 tailed) = .10  Sig 1-tail = .05 • Sig. (2 tailed) = .03  Sig 1-tail = .015

  11. Factors in the Probability of Rejecting H0 For T-tests • The size of the observed difference (produces larger t-observed) 2. The alpha level (need larger t-observed in order to reject null) 3. The use of one or two-tailed tests (two tailed tests make it harder to reject null) 4. The size of the sample (larger N produces larger t-values).

  12. Analysis of Variance • What happens if you have more than two means to compare? • IV (grouping variable) = more than two categories • Examples • Risk level (low medium high) • Race (white, black, native American, other) • DV  Still I/R (mean)

  13. ANOVA = F-TEST • The purpose is very similar to the t-test • HOWEVER • Computes the test statistic “F” instead of “t” • And does this using different logic because you cannot calculate a single distance between three or more means.

  14. ANOVA • Why not use multiple t-tests? • Error compounds at every stage  probability of making an error gets too large • F-test is therefore EXPLORATORY • Independent variable can be any level of measurement • Technically true, but most useful if categories are limited (e.g., 3-5).

  15. Hypothesis testing with ANOVA: • Different route to calculate the test statistic • 2 key concepts for understanding ANOVA: • SSB – between group variation (sum of squares) • SSW – within group variation (sum of squares) • ANOVA compares these 2 type of variance • The greater the SSB relative to the SSW, the more likely that the null hypothesis (of no difference among sample means) can be rejected

  16. Terminology Check • “Sum of Squares” = Sum of Squared Deviations from the Mean = (Xi - X)2 • Variance = sum of squares divided by sample size =  (Xi - X)2 = Mean Square N • Standard Deviation = the square root of the variance = s • ALL INDICATE LEVEL OF “DISPERSION”

  17. The F Ratio • Indicates the variance between the groups, relative to variance within the groups F = Mean square (variance) between Mean square (variance) within • Between-group variance tells us how different the groups are from each other • Within-group variance tells us how different or alike the cases are as a whole sample

  18. ANOVA • Example • Recidivism, measured as mean # of crimes committed in the year following release from custody: • 90 individuals randomly receive 1of the following sentences: • Prison (mean = 3.4) • Split sentence: prison & probation (mean = 2.5) • Probation only (mean = 2.9) • These groups have different means, but ANOVA tells you whether they are statistically significant – bigger than they would be due to chance alone

  19. # of New Offenses: Demo ofBetween & Within Group Variance 2.0 2.5 3.0 3.5 4.0 GREEN: PROBATION (mean = 2.9)

  20. # of New Offenses: Demo ofBetween & Within Group Variance 2.0 2.5 3.0 3.5 4.0 GREEN: PROBATION (mean = 2.9) BLUE: SPLIT SENTENCE (mean = 2.5)

  21. # of New Offenses: Demo ofBetween & Within Group Variance 2.0 2.5 3.0 3.5 4.0 GREEN: PROBATION (mean = 2.9) BLUE: SPLIT SENTENCE (mean = 2.5) RED: PRISON (mean = 3.4)

  22. # of New Offenses: What would less “Within group variation” look like? 2.0 2.5 3.0 3.5 4.0 GREEN: PROBATION (mean = 2.9) BLUE: SPLIT SENTENCE (mean = 2.5) RED: PRISON (mean = 3.4)

  23. ANOVA • Example, continued • Differences (variance) between groups is also called “explained variance” (explained by the sentence different groups received). • Differences within groups (how much individuals within the same group vary) is referred to as “unexplained variance” • Differences among individuals in the same group can’t be explained by the different “treatment” (e.g., type of sentence)

  24. F STATISTIC • When there is more within-group variance than between-group variance, we are essentially saying that there is more unexplained than explained variance • In this situation, we always fail to reject the null hypothesis • This is the reason the F(critical) table (Healey Appendix D) has no values <1

  25. SPSS EXAMPLE • Example: • 1994 county-level data (N=295) • Sentencing outcomes (prison versus other [jail or noncustodial sanction]) for convicted felons • Breakdown of counties by region:

  26. SPSS EXAMPLE • Question: Is there a regional difference in the percentage of felons receiving a prison sentence? • Null hypothesis (H0): • There is no difference across regions in the mean percentage of felons receiving a prison sentence. • Mean percents by region:

  27. SPSS EXAMPLE • These results show that we can reject the null hypothesis that there is no regional difference among the 4 sample means • The differences between the samples are large enough to reject Ho • The F statistic tells you there is almost 20 X more between group variance than within group variance • The number under “Sig.” is the exact probability of obtaining this F by chance A.K.A. “VARIANCE”

  28. ANOVA: Post hoc tests • The ANOVA test is exploratory • ONLY tells you there are sig. differences between means, but not WHICH means • Post hoc (“after the fact”) • Use when F statistic is significant • Run in SPSS to determine which means (of the 3+) are significantly different

  29. OUTPUT: POST HOC TEST • This post hoc test shows that 5 of the 6 mean differences are statistically significant (at the alpha =.05 level) • (numbers with same colors highlight duplicate comparisons) • p value (info under in “Sig.” column) tells us whether the difference between a given pair of means is statistically significant

  30. ANOVA in SPSS • STEPS TO GET THE CORRECT OUTPUT… • ANALYZE  COMPARE MEANS  ONE-WAY ANOVA • INSERT… • INDEPENDENT VARIABLE IN BOX LABELED “FACTOR:” • DEPENDENT VARIABLE IN THE BOX LABELED “DEPENDENT LIST:” • CLICK ON “POST HOC” AND CHOOSE “LSD” • CLICK ON “OPTIONS” AND CHOOSE “DESCRIPTIVE” • YOU CAN IGNORE THE LAST TABLE (HEADED “Homogenous Subsets”) THAT THIS PROCEDURE WILL GIVE YOU

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