19_t-tests__ANOVA__etc

19_t-tests__ANOVA__etc - Hypothesis Testing in Other...

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Unformatted text preview: Hypothesis Testing in Other Contexts Hypothesis Hypothesis testing Hypothesis We have discussed: Relationships b/w categorical vars (chi2) Test Mean or Proportion equals some value (z) Test Difference of Means (z) Hypothesis Testing in Other Contexts: Difference of means if small sample (t) Test difference of means across multiple groups Test (ANOVA) (ANOVA) Significance Test of r Some notes Some Technically, because we do not know the Technically, true population s.e., we should be using the t-test instead of the z-test the But t-test will only produce different results for But small (N<30) samples small Assumptions for t-test: large samples or Assumptions sample from approximately normal distributions, population variances of two samples are equal or sample sizes are roughly equal roughly T-test formula T-test t = M1 – M 2 s.e. s.e. Where s.e. = sqrt(s12/ n1 + s22 / n2 ) t-test example -- Men feel more favorable toward Sarah Palin than women. Sarah Scores are the favorability of Palin on 1-10 scale Men Women 5 6 7 8 10 10 5 6 9 4 4 5 6 5 8 9 4 3 2 4 Mean = 7 S.D. = 2.05 Mean = 5 S.D. = 2.05 t-test example t-test t = 7 - 5 = 3.08 .65 .65 P-value P-value Compare the t-score to value determined by Compare the sample size (N) and level of significance significance If t-score is greater than a critical value, we If critical we reject the null reject Degrees of freedom (df) = n1 + n2 -1 = N - 1 Table I. Student's t distribution critical values Table Probability in TWO-TAILED test (halve for one-tailed) N-1 1 2 3 4 5 6 7 8 9 10 10 11 11 12 12 13 13 14 14 15 15 0.1 0.1 6.134 2.920 2.920 2.353 2.132 2.015 1.943 1.895 1.860 1.833 1.812 1.796 1.782 1.771 1.761 1.753 0.05 12.706 4.303 4.303 3.182 2.776 2.571 2.447 2.447 2.365 2.306 2.262 2.228 2.201 2.179 2.160 2.145 2.131 0.01 63.637 9.925 5.841 4.604 4.032 3.707 3.499 3.355 3.250 3.169 3.106 3.055 3.012 2.977 2.947 0.005 127.320 14.089 7.453 5.598 4.773 4.317 4.020 3.833 3.690 3.581 3.497 3.428 3.372 3.326 3.286 0.001 0.001 636.620 636.620 31.598 31.598 12.924 12.924 8.610 8.610 6.869 6.869 5.959 5.959 5.408 5.408 5.041 5.041 4.781 4.781 4.537 4.537 4.437 4.437 4.318 4.318 4.221 4.221 4.140 4.140 4.073 4.073 N-1 0.1 0.1 19 1.729 0.05 2.093 0.01 0.01 2.861 0.001 3.883 Our t-statistic was 3.08, our df = 20- 1 = 19, and our level of Our signficance is .05, two-tailed signficance We can reject the null, because our test statistic is greater than the We critical value critical Comparing Z and t -distributions Comparing Shape of t-distribution changes for Shape different size samples, so different pdifferent values for different values of N-1 For smaller samples, t-distribution has For “fatter tails” “fatter For larger samples, the two look similar T-distribution & Z-distribution T-distribution Analysis of Variance (ANOVA) (ANOVA) One-way ANOVA examines if differences One-way in means among two or more groups is two result of sampling error or represents a real difference among the groups. real Uses F-test, which compares variances Uses among groups. among Variance and ANOVA Variance Variance is divided into: within group variance--due to error (chance) Among group variance--due to treatment Variance evaluated using F-test, which Variance compares variance due to treatment with error variance. variance. The resulting statistic is compared to a critical The value to determine if the differences in group means are due to chance or an impact of the treatment treatment Distribution of Variance Distribution Within Variance Within Variance Within Variance Among Variance F-test F-test Basic concept for F-test is: F = SS among /(K-1) = MS between SS MS SS within/(N-K) MS within SS SS = sum of the squares; (x-M)2 K = number of groups N = total cases MS = Mean square K-1 & N-K are degrees of freedom ANOVA Intuition ANOVA http://www.psych.utah.edu/stat/introstats/anov F Distribution Distribution ANOVA ANOVA The basic formula is for one-way ANOVA. One-way ANOVA involves one treatment, which can have multiple groups (e.g. McCain Ad, Obama Ad, Control Ad). McCain Factorial ANOVA is used with more than one factor, which allows you to test for interactive effects (e.g. McCain Ad, Obama Ad, Control Ad X Positive Version, Negative Version) Negative ANOVA Example -- Do people learn more from a speech or a speech with visuals? from Control 2 1 4 5 3 Mean 3 Grand mean = 6 Just Speech 5 2 7 8 6 6 Speech with visual add-ons visual 9 10 10 9 7 9 ANOVA Example ANOVA ANOVA Table Source Source Among groups Within groups Total SS df SS 90 2 42 12 132 MS 45 3.5 F 12.86 Sig. p < .01 The 12.86 is larger than the F-distribution critical The rejection value of 6.93. rejection ANOVA Example ANOVA The F-test tells us that the differences in The the means among the three groups were not due to chance not The F-test is an omnibus test for The differences among all the groups the Follow-up analysis can tell us which Follow-up differences are due to treatment and not chance chance ANOVA Example ANOVA One follow-up analysis is to use t-test in One evaluating all two-mean comparisons. evaluating ANOVA Example ANOVA 1) 2) 3) Treatment one - control = 6 - 3 = 3 Treatment two - control = 9 - 3 = 6 Treatment two - treatment one = 9 - 6 = 3 Is difference 1) due to chance? A = yes B = no Difference 2)? Difference 3) ? Difference Hypothesis Testing Example Hypothesis The favorability toward the Iraq War is The measure by a Likert Scale using three statements. (3 - 27). statements. The means for the three groups are: Control -- 12 One-sided argument -- 12 Two-sided argument -- 13 Two-sided Hypothesis Testing Example Hypothesis A. B. The critical value for a two-sided F-tests The at p < .05 with df of 2, 597 is 39.5. at The F for the experiment equals 20.56. Can we reject the null hypothesis? Yes No F Distribution Distribution Hypothesis Testing Example Hypothesis We cannot reject the null hypothesis that We attitudes towards the war did not differ among the three conditions. among Association Inference Association Recall Pearson’s r Measures Association between ratio Measures variables (e.g. income and age) Can test if relationship holds in population population Association Inference Association The null hypothesis for Pearson correlation The is a zero correlation in the population is Two factors affect whether a relationship is Two significant--sample size & correlation size significant--sample Association Inference Association The significance of a Pearson correlation is The based on a separate distribution based The degrees of freedom equals N – 2 If the correlation is greater than a critical If value, you reject the null value, If we are doing a two-tailed test or testing a If negative relationship, use absolute value of r Example Example As the readability of blogs declines (index As increases), visits to the blogs will decline. increases), You randomly sample 202 blog sites, visit You them every day for a week and measure the readability of two postings on each & find average for the week. find You also collect the number of visits You during that week. during Example (con) Example R= -.52. With 202 sites, is this correlation due to With sampling error, or does it exist in the population? population? Can we reject the null hypothesis of no Can correlation? correlation? Critical Values for r df 4 6 8 10 12 14 16 18 20 25 30 35 40 45 50 60 70 80 90 10 0 20 0 30 0 40 0 50 0 9 0% .72 9 .62 2 .54 9 .49 7 .45 8 .42 6 .40 0 .37 8 .36 0 .32 3 .29 5 .27 5 .25 7 .24 3 .23 1 .21 1 .19 5 .18 3 .17 3 .16 4 .11 6 .09 5 .08 2 .07 3 95% .81 1 .70 7 .63 2 .57 6 .53 2 .49 7 .46 8 .44 4 .42 3 .38 1 .34 9 .32 5 .30 4 .28 8 .27 3 .25 0 .23 2 .21 7 .20 5 .19 5 .13 8 .11 3 .09 8 .08 8 9 8% .88 2 .78 9 .71 6 .65 8 .61 2 .57 4 .54 2 .51 6 .49 2 .44 5 .40 9 .38 1 .35 8 .33 8 .32 2 .29 5 .27 4 .25 6 .24 2 .23 0 .16 4 .13 4 .11 6 .10 4 9 9% .91 7 .83 4 .76 5 .70 8 .66 1 .62 3 .59 0 .56 1 .53 7 .48 7 .44 9 .41 8 .39 3 .37 2 .35 4 .32 5 .30 2 .28 3 .26 7 .25 4 .18 1 .14 8 .12 8 .11 5 Association Inference Association Critical correlation value, 2-sided test, Critical p < .01, df = 200 equals .181. .01, We can reject the null hypothesis Questions? ...
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