Parametric Analysis of Variance

Transcription

1 Parametric Analysis of Variance

2 The same principles used in the two-sample means test can be applied to more than two sample means. In these cases we call the test analysis of variance. The null hypothesis for analysis of variance (AOV or ANOVA) is: H 0 : μ 1 μ μ k And this may be applied over both time and space.

3 Assumptions of analysis of variance testing: 1. Observations between and within samples are random and independent (methodological issue).. The observations in each category are normally distributed (run normality tests on each group). 3. The population variances are assumed to be equal (homogeneity of variances test).

4 j Generic Data Set i Group1 Group Group3 Obs 1 X 1,1 X,1 X 3,1 Obs X 1, X, X 3, Obs 3 X 1,3 X,3 X 3,3 Obs 4 X 1,4 X,4 X 3,4 Obs 5 X 1,5 X,5 X 3,5 Note that i denotes rows and j denotes columns.

5 AOV: Compares the variation within the groups (columns) to the variation among the group means. If the variation among the group means is much greater than within the groups, reject H 0. If the variation among the group means is not much greater than within the groups, accept H 0.

6 There are 3 pieces of information that need to be determined to perform AOV: TSS (total sum of squares) the sum of the squared deviations of the observation from the overall mean. BSS (between sum of squares) the sum of the squared deviations between the groups. WSS (within sum of squares) the sum of the squared deviations with the groups.

7 We will use this set of equations because they are easier to calculate. TSS k i 1 n i j 1 X ij C where C ( X ij ) N BSS n i i 1 n i j 1 X n i ij C WSS TSS BSS

8 We test for differences among the groups using the F test. The standard F test is simply a ratio of variances. Here the F test is slightly modified to take into account: 1. The number of groups.. Differences in the number of group members.

9 The F statistic is calculated as: F BSS k 1 WSS N k where k is the number of groups and N is the total sample size. This is basically the sample size weighted ratio of the within group variation to the between group variation.

10 There are types of degrees of freedom for the F statistic: k 1 (where k is the number of columns) for the numerator. N k (where N is the total sample size) for the denominator. Note that the table from the book is different than the web table. Book p values are < or > a set alpha level. Web table p values can be a range.

11 A few words on Sum of Squares In all cases, the sum of squares is a measure of total variation from the mean in a data set. If the sum of squares is large, then the data set has a lot of variation. This makes it more difficult to reject H o (find a difference). If the sum of squares is small, then the data set has little variation. This makes it easier to reject H o (find a difference). Once the sum of squares has been determined there are many ways of using the information.

12 In AOV the sum of squares is partitioned into 3 distinct types. This is because we are working with subsets of the total data set (3+ groups). It can be thought of as performing multiple and simultaneous difference in means tests (t-tests).

13 Understanding the role of the total sum of squares (TSS) is straightforward: it is the total amount of variation in the data set. TSS i j ( X ij X ij ) is equivalent to TSS k i 1 n i j 1 X ij C where C ( X ij ) N

14 For TSS the data are pooled together.

15 The between sum of squares (BSS) is the treatment variance. We want to compare these to each other. j ij j j X X n BSS ) ( C n X BSS i i n i i n j ij 1 1 is equivalent to

16 We are examining the distance between the means of each group (A and B).

17 The within sum of squares (WSS) can be thought of as the error variance. We want to factor out variation within the groups. WSS i j ( X ij X j ) is equivalent to WSS TSS BSS

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19 The F test is simply the ratio between the error variance (within groups) and the treatment variance (between groups). F BSS k 1 WSS N k IF the ratio is small then the difference is also small (accept H o ). IF the ratio is large then the difference is also large (reject H o ).

20 Tarascan Village Percent Population Employed Mountain Group Meseta Group Lake Group Mountain Meseta Lake We will drop the towns along the highways since we are interested only in rural towns.

21 Lake Meseta Mountain Quiroga Arantepacua Ahuiran San Andres 54.1 Comachuen 36.7 Angahuan 47.1 San Jeronimo La Mojonera Charapan 45.9 Erongaricuaro Nahuatzen Cocucho 4.41 Ihuatzio Pichataro Corupo 38.9 Jaracuaro Quinceo 3.64 Nurio Puacuaro 50.5 San Isidro 44.3 Pomacuaran Santa Fe San Juan 3.76 San Felipe Uricho 33. Sevina San Lorenzo Zurumutaro Tingambato 44.7 Urapicho 9.90 Tzintzuntzan Turicuaro Ajuno 37.5

22 Normality Tests by Group Ho: Village economic activity measurements are not significantly different than normal qlake 3.61 Meseta q q Mountain n Lake 1 q critical Accept Ho. n Meseta 11 q critical Accept Ho. n Mountain 10 q critical Accept Ho. All village groups are not significantly different than normal (w/s q Lake 3.61, p > 0.05, q Meseta.86, p > 0.05, q Mountain.97, p > 0.05)

23 Equality of Variances Test (Hartley Test 1 ) Ho: The group variances are not significantly different. F Max df df n F F lake max 11, critical n Max Min max 1 3 s s 1, k Group variances: Lake Meseta Mountain The group variances are not significantly different (F 3.6, p > 0.05). 1 SPSS uses the less conservative Levene s Test.

24 Use next lower value.

25 The Hartley test is VERY sensitive to the normality assumption. Even small deviations from normality can influence the results. The test works best when the group sample sizes are equal or roughly equal. If very unequal use the group with the largest sample size (rather than largest variance) for determining n. If in doubt, double check using the SPSS Levene s test.

26 Hypotheses H o : There is no significant difference in the percent working population among the three Tarascan village groups. H a : There is a significant difference in the percent working population among the three Tarascan village groups. Use between for two-group testing and among for 3+ group testing.

27 Lake Meseta Mountain Quiroga Arantepacua Ahuiran San Andres 54.1 Comachuen 36.7 Angahuan 47.1 San Jeronimo La Mojonera Charapan 45.9 Erongaricuaro Nahuatzen Cocucho 4.41 Ihuatzio Pichataro Corupo 38.9 Jaracuaro Quinceo 3.64 Nurio Puacuaro 50.5 San Isidro 44.3 Pomacuaran Santa Fe San Juan 3.76 San Felipe Uricho 33. Sevina San Lorenzo Zurumutaro Tingambato 44.7 Urapicho 9.90 Tzintzuntzan Turicuaro Ajuno 37.5 Column Sums n 1 1 n 11 n 3 10 n total 33

28 TSS k i 1 n i j 1 X ij C where C ( X ij ) N Lake x Meseta x Mountain x Quiroga Arantepacua Ahuiran San Andres Comachuen Angahuan San Jeronimo La Mojonera Charapan Erongaricuaro Nahuatzen Cocucho Ihuatzio Pichataro Corupo Jaracuaro Quinceo Nurio Puacuaro San Isidro Pomacuaran Santa Fe San Juan San Felipe Uricho Sevina San Lorenzo Zurumutaro Tingambato Urapicho Tzintzuntzan Turicuaro Ajuno Column Sums Sum of the observations Sum of the squared observations Total observation sum ( X ij ) Total squared sum k i 1 n i j 1 X ij

29 TSS k i 1 n i j 1 X ij C where C ( X ij ) N C (1419.) TSS

30 BSS n i i 1 n i j 1 X n i ij C Lake n1 x Meseta n11 x Mountain n10 x Quiroga Arantepacua Ahuiran San Andres Comachuen Angahuan San Jeronimo La Mojonera Charapan Erongaricuaro Nahuatzen Cocucho Ihuatzio Pichataro Corupo Jaracuaro Quinceo Nurio Puacuaro San Isidro Pomacuaran Santa Fe San Juan San Felipe Uricho Sevina San Lorenzo Zurumutaro Tingambato Urapicho Tzintzuntzan Turicuaro Ajuno Column Sums BSS (593.5) 1 + (49.8) 11 + (395.9)

31 TSS BSS WSS Only used to calculate the WSS F BSS k 1 WSS N k F df (k-1) (3-1), (n-k) (33-3) 30

32 To get the probability range on this table, read DOWN the column. F 5.6

33 Since 5.6 > 3.3 reject H o There is a significant difference in the percent employed population among the three groups of Tarascan villages (F 5.6, 0.01 > p > 0.005).

34 Test of Homogeneity of Variances PctAct Levene Statistic df1 df Sig ANOVA PctAct Between Groups W ithin Groups Total Sum of Squares df Mean Square F Sig Multiple Comparisons Dependent Variable: PctAct LSD (I) Group 1 3 (J) Group Me an Difference 95% Confidence Interval (I-J) Std. Error Sig. Lower Bound Upper Bound * * * *. The mean difference is s ignificant at the.05 level *

35 Comparing our results to those of SPSS: SPSS Us TSS BSS WSS F p

36 Multiple Comparison (AOV Post Hoc) Tests Available in SPSS to compare all group pairs. Used to determine which means are significantly different. Similar to (but not exactly like) performing two group means tests while holding the other group(s) constant. The post hoc test listed below are useful for non-paired comparisons and non treatment-control comparisons. For treatment-control (e.g. before- after) comparisons, use the Dunnet test.

37 LSD (Fisher test) The Least Significant Difference test assumes that if the null hypothesis is incorrect, as indicated by a significant F-test, Type I errors are not really possible. LSD test has been criticized for not sufficiently controlling for Type I errors. Although this is the first test on the list in SPSS, it is probably not the best.

38 Bonferroni Probably the most commonly used post hoc test because it is flexible, simple to compute, and can be used with any type of statistical test. The traditional Bonferroni tends to lack power. The Bonferroni overcorrects for Type I error.

39 Sidak A relatively simple modification of the Bonferroni formula that would have less of an impact on statistical power but retain much of the flexibility of the Bonferroni method. This approach is convenient but has not received any systematic study. It is unlikely that a single, simple correction will result in the most efficient balance of Type I and Type II errors.

40 Tukey (Tukey s HSD) The HSD (honest significant difference) has greater power than the other tests under most circumstances. Requires equal sample sizes. If sample sizes are unequal, the Tukey-Kramer test is used by SPSS which computes the harmonic mean of the sample sizes for each group, slightly lowering the power of the test.

41 Games-Howell This test can be used when variances are unequal and also takes into account unequal group sizes. Severely unequal variances can lead to increased Type I error, and with smaller sample sizes, more moderate differences in group variance can also lead to increases in Type I error. This test appears to do better than the Tukey HSD if variances are very unequal (or moderately so in combination with small sample size) or can be used if the sample size per cell is very small (e.g., <6). This is probably the most reliable post hoc test, but also the most conservative.

42 Probabilities from 5 Post Hoc Tests Group Games- Comparisons Howell Tukey HSD LSD Bonferroni Sidak Note that the differences are slight, except for the Games- Howell test.

43 Texas Death Row Inmates (016) Race/ Ethnicity Age Race/ Ethnicity Age Race/ Ethnicity Age White 44 Black 9 Hispanic 6 White 37 Black 36 Hispanic 3 White 7 Black 19 Hispanic 0 White 8 Black 3 Hispanic 5 White 40 Black 5 Hispanic 9 White 18 Black 19 Hispanic 36 White 19 Black 8 White 41 Black Black 3 Are the ages (at time of offense) of Texas death row inmates by race/ethnicity different?