Statistiek II. John Nerbonne. October 1, Dept of Information Science


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1 Dept of Information Science October 1, 2010
2 Course outline 1 Oneway ANOVA. 2 Factorial ANOVA. 3 Repeated measures ANOVA. 4 Correlation and regression. 5 Multiple regression. 6 Logistic regression.
3 Lecture outline Today: Oneway ANOVA 1 General motivation 2 F test and F distribution 3 ANOVA example 4 The logic of ANOVA Short break 5 ANOVA calculations 6 Posthoc tests
4 What s ANalysis Of VAriance (ANOVA)? Most popular statistical test for numerical data Generalized ttest Compares means of more than two groups Fairly robust Based on F distribution compares variances (between groups and within groups) Two basic versions: a Oneway (or single) ANOVA: compare groups along one dimension, e.g., grade point average by school class b Nway (or factorial) ANOVA: compare groups along 2 dimensions, e.g., grade point average by school class and gender
5 Typical applications Oneway ANOVA: Compare time needed for lexical recognition in 1. healthy adults 2. patients with Wernicke s aphasia 3. patients with Broca s aphasia Factorial ANOVA: Compare lexical recognition time in male and female in the same three groups
6 Comparing multiple means For two groups: use ttest Note: testing for pvalue of 0.05 shows significance 1 time in 20 if there is no difference in population mean (effect of chance) But suppose there are 7 groups, i.e., we test ( 7 2) = 21 pairs of groups Caution: several tests (on the same data) run the risk of finding significance through sheer chance
7 Multiple comparison problem Example: Suppose you run k = 3 tests, always seeking a result significant at α = 0.05 probability of getting at least one false positive is given by: α FW = 1 P(zero false positive results) = 1 (1 α) k = 1 (1 0.05) 3 = 1 (0.95) 3 = Hence, with only 3 pairwise tests, the chance of committing type I error almost 15% (and 66% for 21 tests!) α FW called Bonferroni familywise αlevel
8 Bonferroni correction for multiple comparisons To guarantee a familywise αlevel of 0.05, divide α by number of tests. Example: 0.05/3 (= α/# tests) = (note: ) set α = (= Bonferronicorrected αlevel) If pvalue is less than the Bonferronicorrected target α: reject the null hypothesis. If pvalue greater than the Bonferronicorrected target α: do not reject the null hypothesis.
9 Analysis of variance ANOVA automatically corrects for looking at several relationships (like Bonferroni correction) Based on F distribution: Moore & McCabe, 7.3, pp Measures the difference between two variances (variance σ 2 ) F = s2 1 s 2 2 always positive since variances are positive two degrees of freedom interesting, one for s 1, one for s 2
10 F test vs. F distribution F value: F = s2 1 s 2 2 F values used in F test (Fisher s test) H 0 : samples are from same distribution (s 1 = s 2 ) H a : samples are from different distributions (s 1 s 2 )  value near 1 indicates same variance  value near 0 or + indicates difference in variance F test very sensitive to deviations from normal ANOVA uses F distribution, but is different: ANOVA F test!
11 F distribution Critical area for F distribution at p = 0.05 (df: 12,10) Probability density rejection region acceptance region rejection region Note the symmetry: P( s2 1 s 2 2 F value < x) = P( s2 2 s 2 1 (because y < x 1 y > 1 x for x, y R + ) > 1 x )
12 F test Example: height group sample mean standard size deviation boys cm 6cm girls 9 168cm 4cm Is the difference in standard deviation significant? Examine F = s2 boys s 2 girls Degrees of freedom: df boys = 16 1 df girls = 9 1
13 F test critical area (for twotailed test with α = 0.05) P(F (15, 8) > x) = α 2 = P(F (15, 8) < x) = P(F (15, 8) < 4.1) = Moore & McCabe, Table E, p. 706 P(F (15, 8) < x) = P(F (8, 15) > x ) = where x = 1 x P(F (8, 15) > 3.2) = (tables) P(F (15, 8) < ) = P(F (15, 8) < 0.31) = (no values directly for P(F (df 1, df 2 ) > x)) Reject H 0 if F < 0.31 or F > 4.1 Here, F = = 2.25 (hence no evidence of difference in distributions)
14 ANOVA Analysis of Variance (ANOVA) most popular statistical test for numerical data several types  single, oneway  factorial, two, three,..., nway  single/factorial repeated measures examines variation  betweengroups gender, age, etc.  withingroups overall automatically corrects for looking at several relationships (like Bonferroni correction) uses F distribution, where F (n, m) fixes n typically at the number of groups (minus 1), m at the number of subjects, i.e., data points (minus number of groups)
15 Detailed example: oneway ANOVA Question: Are exam grades of four groups of foreign students Nederlands voor anderstaligen the same? More exactly, are the four averages the same? H 0 : µ 1 = µ 2 = µ 3 = µ 4 H a : µ 1 µ 2 or µ 1 µ 3... or µ 3 µ 4 Alternative hypothesis: at least one group has a different mean For the question of whether any particular pair is different, the ttest is appropriate. For testing whether all language groups are the same, pairwise ttests exaggerate differences (increase the chance of type I error) We therefore want to apply oneway ANOVA
16 Data: Dutch proficiency of foreigners Four groups of ten students each: Group Europe America Africa Asia Mean Samp. SD Samp. Variance
17 ANOVA conditions ANOVA assumptions: Normal distribution per subgroup Same variance in subgroups: least SD > onehalf of largest SD independent observations: watch out for testretest situations! Check differences in SD s! (some SPSS computing)
18 ANOVA conditions Assumption: normal distribution per group, check with normal quantile plot, e.g., for Europeans below (repeat for every group) Normal QQ plot of toets.nl voor anderstalige
19 Visualizing ANOVA data Is there a significant difference in the means (of the groups being contrasted)? Take care that boxplots sketch medians not means.
20 Sketch of ANOVA Group Eur. Amer. Africa Asia x 1j x 2j x 3j x 4j x 1 x 2 x 3 x 4 Notation: Group index: i {1, 2, 3, 4} Sample index: j N i = size of group i Data point x ij : ith group, jth observation Number of groups: I = 4 Total mean: x Group mean: x i For any data point x ij : (x ij x) = (x i x) + (x ij x i ) total residue = group diff. + error ANOVA question: does group membership influence the response variable?
21 Two variances Reminder of high school algebra: (a + b) 2 = a 2 + b 2 + 2ab ab a 2 a b 2 ab b
22 Two variances Data point x ij : (x ij x) = (x i x) + (x ij x i ) Want sum of squared deviates for each group: (x ij x) 2 = (x i x) 2 + (x ij x i ) 2 + 2(x i x)(x ij x i ) Sum over elements in ith group: N i (x ij x) 2 = j=1 N i N i (x i x) 2 + (x ij x i ) 2 + 2(x i x)(x ij x i ) j=1 j=1 j=1 N i
23 Two variances Note that this term must be zero: Because: N i 2(x i x)(x ij x i ) j=1 (a) (b) N i 2(x i x)(x ij x i ) = 2(x i x) (x ij x i ) j=1 N i (x ij x i ) = 0 x i = j=1 N i j=1 } {{ } Ni j=1 x ij 0 N i
24 Two variances So we have: N i j=1 (x ij x) 2 = N i N i (x i x) 2 + (x ij x i ) 2 j=1 j=1 N i (+ 2(x i x)(x ij x i ) = 0) j=1 Therefore: N i j=1 (x ij x) 2 = N i N i (x i x) 2 + (x ij x i ) 2 j=1 j=1 And finally we can sum over all groups: I N i I N i I N i (x ij x) 2 = (x i x) 2 + (x ij x i ) 2 i=1 j=1 i=1 j=1 i=1 j=1
25 ANOVA terminology (x ij x) = (x i x) + (x ij x i ) total residue = group diff. + error I N i I I N i (x ij x) 2 = N i (x i x) 2 + (x ij x i ) 2 i=1 j=1 i=1 i=1 j=1 SST SSG SSE Total Sum of Squares = Group Sum of Squares + Error Sum of Squares (n 1) = (I 1) + (n I ) DFT DFG DFE Total Degrees of Freedom = Group Degrees of Freedom + Error Degrees of Freedom
26 Variances are mean squared differences to the mean Note that SST/DFT: P I P Ni i=1 j=1 (x ij x) 2 n 1 is a variance, and likewise SSG/DFG: labelled MSG ( Mean square between groups ), and SSE/DFE: labelled MSE ( Mean square error or sometimes Mean square within groups ) In ANOVA, we compare MSG (variance between groups) and MSE (variance within groups), i.e. we measure F = MSG MSE If this F value is large, differences between groups overshadow differences within groups.
27 Two variances 1) Estimate the pooled variance of the population (MSE): MSE = DFE SSE PI P Ni = i=1 j=1 (x ij x i ) 2 n I equiv = P I i=1 DF i s 2 i P I i=1 DF i In our example (Nederlands for anderstaligen): P I i=1 DF i s 2 i P I i=1 DF i = (N1 1)s2 1 + (N 3 1)s (N 3 1)s (N 4 1)s 2 4 (N 1 1) + (N 3 1) + (N 3 1) + (N 4 1) = = = Estimates the variance in groups (using DF), aka withingroups estimate of variance
28 Two variances 2) Estimate the betweengroups variance of the population (MSG): MSG = DFG SSG P = I i=1 N i (x i x) 2 I 1 In our example (Nederlands for anderstaligen): We had 4 group means: 25.0, 21.9, 23.1, 21.3, grand mean: 22.8 MSG = 10 (( )2 +( ) 2 +( ) 2 +( ) 2 ) 4 1 = 26.6 The betweengroups variance (MSG) is an aggregate estimate of the degree to which the four sample means differ from one another
29 Interpreting estimates with F scores If H 0 is true, then we have two variances: Betweengroups estimate: s 2 bg = Withingroups estimate: s 2 wg = and and their ratio s 2 bg s 2 wg follows an F distribution with: (# groups 1) = 3 degrees of freedom for s bg 2 and (# observations # groups) = 36 degrees of freedom for swg 2 In our example: F (3, 36) = = 0.55 P(F (3, 40) > 2.84) = 0.05 (see tables), so there is no evidence of nonuniform behavior
30 SPSS summary No evidence of nonuniform behavior
31 Other questions ANOVA H 0 : µ 1 = µ 2 =... = µ n But sometimes particular contrasts are important e.g., are Europeans better (in learning Dutch)? Distinguish (in reporting results): prior contrasts questions asked before data is collected and analyzed post hoc (posterior) questions questions asked after data collection and analysis datasnooping is exploratory, cannot contribute to hypothesis testing
32 Prior contrasts Questions asked before data collection and analysis e.g., are Europeans better (in learning Dutch)? Another way of putting this: H 0 : µ Eur = 1 3 (µ Am + µ Afr + µ Asia ) H a : µ Eur 1 3 (µ Am + µ Afr + µ Asia ) Reformulation (SPSS requires this): H 0 : 0 = µ Eur µ Am µ Afr µ Asia
33 Prior contrasts in SPSS Mean of every group gets a coefficient Sum of coefficients is 0 A ttest is carried out and twotailed pvalue is reported (as usual): No significant difference here (of course) Note: prior contrasts are legitimate as hypothesis tests as long as they are formulated before data collection and analysis
34 Posthoc questions Assume H 0 is rejected: which means are distinct? Datasnooping problem: in a large set, some distinctions are likely to be statistically significant But we can still look (we just cannot claim to have tested the hypothesis) We are asking whether m i m j is significantly larger, we apply a variant of the ttest The relevant sd is MSE n (differences among scores), but there is a correction since we re looking at a proportion of the scores in any one comparison
35 SD in posthoc ANOVA questions Standard deviation (among differences in groups i and j): sd = t = MSE N i +N j N = = 4.9 x i x j sd q 1 N i + 1 N j The critical tvalue is calculated as p c where p is the desired significance level and c is the number of comparisons. For pairwise comparisons: c = ( ) I 2
36 Posthoc questions in SPSS SPSS posthoc Bonferroni searches among all groupings for statistically significant ones But in this case there are none (of course)
37 How to win at ANOVA Note the ways in which the F ratio increases (i.e., becomes more significant): F = MSG MSE 1. MSG increases: differences in means between groups grow larger 2. MSE decreases: overall variation within groups grows smaller
38 Two models for grouped data x ij = µ + ɛ ij x ij = µ + α i + ɛ ij First model: no group effect each data point represents error (ɛ) around a mean (µ) Second model: real group effect each data point represents error (ɛ) around an overall mean (µ), combined with a group adjustment (α i ) ANOVA asks: is there sufficient evidence for α i?
39 Next week Next week: factorial ANOVA
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