2. RELATIONSHIP BETWEEN A QUALITATIVE AND A QUANTITATIVE VARIABLE

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1 7/09/06. RELATIONHIP BETWEEN A QUALITATIVE AND A QUANTITATIVE VARIABLE Design and Data Analysis in Psychology II usana anduvete Chaves alvadorchacón Moscoso. INTRODUCTION You may examine gender differences in average salary or racial (white versus black) differences in average annual income. The variable (salary) to be tested should be interval or ratio (quantitative), whereas the gender or race variable should be binary (qualitative).

2 7/09/06. INTRODUCTION These analyses are used to compare group means (independent or dependent groups). The term independentis used because groups are conformed randomly (independent samples). Two separate sets of independent and identically distributed samples are obtained. 3. INTRODUCTION The term dependent is used because groups are paired with respect to some measure (dependent samples). Dependent samples (or "paired") comparison consist of a sample of matched pairs of similar units, or one group of units that has been tested twice. 4

3 7/09/06. INTRODUCTION A typical example of the repeated measures would be where subjects are tested prior to a treatment, say for high blood pressure, and the same subjects are tested again after treatment with a blood-pressure lowering medication. 5. INTRODUCTION Three steps to carry out:. Testing statistical assumptions.. tatistical tests of significance. 3. tatistical power. 6 3

4 7/09/06. TEP. TETING AUMPTION Quantitative variable: Interval or ratio variable. Normal distribution: the populations from which the samples are selected must be normal (Tests for Normality: Kolmogorov- mirnov D + Histogram of predicted Z scores). 7. TEP. TETING AUMPTION The Central Limit Theorem says, however, that the distributions of y and y are approximately normal when N is large. In practice, when n + n 30, you do not need to worry too much about the normality assumption. 8 4

5 7/09/06. TEP. TETING AUMPTION Independence: the observations within each sample must be independent (Durbin-Watson D + scatter plot X-e). Homoscedasticity(homogeneity of variance): the populations from which the samples are selected must have equal variances. In this kind of studies, it is the most important assumption to take into account (Levene s test, F Max + scatter plot y -absolute errors). 9. TEP. TETING AUMPTION To test homoscedasticity: - Levene s test (P). - F-Max test: Fmax ( l argest) ( smallest) F F (α, k, n ) Null hypothesis is accepted. The homogeneity of variance assumption has been satisfied. F > F (α, k, n ) Null hypothesis is rejected. The homogeneity of variance assumption has been violated. * k number of groups (number of variances) n number of participants in each group (same size is assumed) 0 5

6 7/09/06. TEP. TETING AUMPTION: EXAMPLE Test the homogeneity of variance assumption in the following data (A nationality; Y level of depression; α0.05): a : panish a : Japanese TEP. TETING AUMPTION: EXAMPLE a a : a a Σ5 Σ75 Σ93 Σ58 Y Y Y Y N Y N Y Y Y N Y Y N

7 7/09/06. TEP. TETING AUMPTION: EXAMPLE Fmax ( l arg est) ( smallest).85 F (α, k, n ) F (0.05,, 0 ) 4.43 F F (α, k, n ) Null hypothesis is accepted. The homogeneity of variance assumption has been satisfied tep. tatistical Tests of ignificance 3.. tep. Non-parametric Tests of ignificance: assumptions are rejected 3.. tep. Parametric tests of significance: assumptions are accepted 4 7

8 7/09/ TEP : NON-PARAMETRIC TET 3... Wilcoxon T test. -When the population cannot be assumed to be normally distributed. - Quantitative dependent variable. - Two paired groups. - Formula: T + R i T R i T the smallest of these two rank sums 5 Conclusions: 3... Wilcoxon T test T T (α, N) The null hypothesis is accepted. The variables are not related. There are not differences between groups. T < T (α, N) The null hypothesis is rejected. The variables are related. There are differences between groups. * N number of pairs removing those which difference is

9 7/09/ Wilcoxon T test: example In the table below, you can see the punctuation in social skills that a twin who went to the kindergarten (a ) for a course, and the other twin who stayed at home (a ). Are there statistical differences between both groups in social skills? (α0.05). a a Wilcoxon T test: example a a d Rank ign

10 7/09/ Wilcoxon T test: example T T + 36 T 9 R i 9 T (α, N) T (0.05, 9) 5 T > T (α, N) 9 > 5 The null hypothesis is accepted. The variables are not related. There are not statistical differences between groups in social skills Mann-Whitney U - Quantitative or ordinal dependent variable. - Two independent groups. - Calculation: U nn U n n n ( n + ) + n( n + ) + U chosen the smallest one n smallest sample; n largest sample R R 0 0

11 7/09/06 Conclusions: 3... Mann-Whitney U U U (α, n, n) The null hypothesis is accepted. The variables are not related. There are not statistical differences between groups. U < U (α, n, n) The null hypothesis is rejected. The variables are related. There are statistical differences between groups Mann-Whitney U: example We want to study if two different reading methods imply different results in reading qualifications (α0.05). Type of method Qualifications Traditional (a ) New (a )

12 7/09/ Mann-Whitney U: example Type of method Qualifications ΣR Traditional (a ) 80 (4) 85 (5) 5 () 70 (3) 90 (6) 9 New (a ) 95 (8) 00 (9) 93 (7) 0 (0) 45 () 36 *Values into brackets ranks (orders) U U nn n n n ( n + ) + n( n + ) + 5(5 + ) R 5* (5 + ) R 5* U Mann-Whitney U: example U > U (0.05, 5, 5) 4 > The null hypothesis is accepted. The variables are not related. There are not statistical differences in reading qualifications depending on the reading method. 4

13 7/09/ Kruskal Wallis H - Quantitative or ordinal dependent variable. - k independent groups. - Formula: H k R N( N + ) n j j j 3( N + ) 5 Conclusions: Kruskal Wallis H H H (α, k, n, n, n3) The null hypothesis is accepted. The variables are not related. There are not statistical differences between groups. H > H (α, k, n, n, n3) The null hypothesis is rejected. The variables are related. There are statistical differences between groups. 6 3

14 7/09/ Kruskal Wallis H: example Do exist statistical differences in levels of authoritarianism between teachers from three different types of schools? (α0.05) tateschool (a ) Privateschool (a ) tate-subsidized school(a 3 ) Kruskal Wallis H: example tateschool (a ) Privateschool (a ) tate-subsidized school(a 3 ) 96 (4) 8 () 5 (7) 8 (9) 4 (8) 49 (3) 83 (3) 3 (0) 66 (4) 6 () 35 () 47 () 0 (5) 09 (6) ΣR ΣR 37 ΣR 3 46 *Values into brackets ranks (orders) 8 4

15 7/09/ Kruskal Wallis H: example k R H N ( N + ) n () 4(4 ) + 5 j j j 3( N (37) ) (46) 4 3(4 + ) 6.4 H > H (α, k, n, n, n3) 6.4 > The null hypothesis is rejected. The variables are related. There are statistical differences in levels of authoritarianism between teachers from different schools TEP. PARAMETRIC TET OF IGNIFICANCE 3... Linear regression analysis: qualitative independent variable with: -Two groups: lesson 3 (simple linear regression). -More than two groups: lesson 4 (multiple linear regression) T-tests (independent and dependent samples) One-way ANOVA F. 30 5

16 7/09/06 - Two groups T-tests - Quantitative dependent variable (Interval or ratio variable). - Different situations: Independent two-sample t-test. Equal sample sizes, equal variances in the distribution. Equal sample sizes, unequal variances in the distribution. Unequal sample sizes, equal variances in the distribution. Unequal sample sizes, unequal variances in the distribution. 3 Dependent t-test for paired samples T-tests: significance in all cases t > t ( theoretica l ) Null hypothesis is rejected. The model is valid. The slope is statistically different from 0. There is, therefore, relationship between variables. t t( theoretical) Null hypothesis is accepted. The model is not valid. The slope is statistically equal to 0. There is not, therefore, relationship between variables. 3 6

17 7/09/06 t 3... T-tests: two independent samples X X X Equal sample sizes, equal variances X n Where the grand standard deviation or pooled standard deviation is X X X + X The degrees of freedom for this test is N T-tests: two independent samples Equal sample sizes, equal variances: example A researcher wants to study the effect of depriving rats of water, and the time they spend in covering a labyrinth. 60 rats were assigned randomly to -hour deprivation group (sample A) and 4-hour deprivation group (sample B). The 30 rats in group A spent a mean of 5.9 minutes in covering the labyrinth, whereas the 30 rats in group B spent a mean of 6.4 minutes. The standard deviation were 0.9 and 0.3 respectively. Assuming that the two distributions have the same variance, did statistical differences between groups exist? (α0.05) 34 7

18 7/09/ T-tests: two independent samples Equal sample sizes, equal variances: example X t X XX n * XX X + X t > t ( α, N ) t > t (0.05,58) > Null hypothesis is rejected. There were statistical differences between groups T-tests: two independent samples Equal sample sizes, unequal variances X t X X n Where the grand standard deviation or pooled standard deviation is X X X The degrees of freedom for this test is: X + d. f. ( X / n ) ( / n + / n ) /( n ) + ( / n ) /( n ) 36 8

19 7/09/ T-tests: two independent samples Equal sample sizes, unequal variances: example If in the previous example about rats, we assume that the variances of the two distributions were unequal, would statistical differences between variances exist? (α0.05) T-tests: two independent samples Equal sample sizes, unequal variances: example t d. f. ( t > t ( α, d. f.) ( / n + / n ) ( 0.9 / /30) / n ) /( n ) + ( t > t (0.05,58) / n ) /( n 6.579> ) (0.9 /30) /(30 ) + (0.3 /30) /(30 ) Null hypothesis is rejected. There were statistical differences between groups 38 9

20 7/09/06 t where X X 3... T-tests (two independent samples) X X X X + n n Unequal sample sizes, equal variances ( n ) + ( n ) X X n + n The degrees of freedom for this test is n + n T-tests (two independent samples) Unequal sample sizes, equal variances: example A researcher wanted to study the level of relax and the flexibility. 9 gymnasts were assigned randomly to two groups. The first group (experimental) participated in a 4-week training program with relax and flexibility exercises; the second group (control) participated in the same program, but without relax exercises. The measures in flexibility obtained were the following: E: C: Variances are 58.7 in E and 4.67 in C. Populations present equal variances. Are there statistical differences between groups? (α0.05) 40 0

21 7/09/ T-tests (two independent samples) Unequal sample sizes, equal variances t X X E C X X X n n X X E X C + n n E C T-tests (two independent samples) Unequal sample sizes, equal variances X X ( n ) X + ( n ) X ( ) + ( 4 ) n + n X X 4* * t > t( α, n ) > (0.05,7) > n t t Null hypothesis is rejected. There were statistical differences between groups. 4

22 7/09/ T-tests (two independent samples) Unequal sample sizes, unequal variances t X X X X Where the not pooled standard deviation is: d. f. ( / n ) + X X n ( / n + / n ) /( n n ) + ( / n ) /( n ) T-tests (two independent samples) Unequal sample sizes, unequal variances: example upposing that in the previous example about gymnasts, the populations present unequal variances, are there differences between groups? (α0.05) 44

23 7/09/ T-tests (two independent samples) Unequal sample sizes, unequal variances: example t X X X X X X X X n + n T-tests (two independent samples) d. f. ( d. f. (58.7 / 5) / Unequal sample sizes, unequal variances: example ( / n + / n ) n ) /( n ) + ( / n) /( n ) ( 58.7 / / 4) /(5 ) + (4.67 / 4) ( ) d. f. (.74) / 4 + (0.47) / 3 /(4 ) d. f t > t t > t ( α, d. f ) (0.05,6.95) > / / 3 Null hypothesis is rejected. There were statistical differences between groups. 46 3

24 7/09/ T-tests (two dependent samples) Dependent t-test for paired samples D µ 0 t D D D D n j / n ( D D ) j + n n + n The constant µ 0 is non-zero if you want to test whether the average of the difference is significantly different from µ 0. The degree of freedom used is n T-tests (two dependent samples) Dependent t-test for paired samples A researcher wanted to study the level of relax and the flexibility. 9 gymnasts were measured before and after participating in a 4-week training program with relax and flexibility exercises. The measures in flexibility obtained were the following: Before (B): After (A) : Are there statistically differences between the measures obtained before and after the program? (α0.05) 48 4

25 7/09/ T-tests (two dependent samples) Dependent t-test for paired samples B A D j D j D ( D j D) t D D D 3... T-tests (two dependent samples) D µ 0 D D n j / n Dependent t-test for paired samples / ( D D ) j + n * 0.4 n n t > t(, n ) t > t(0.05,4) 3.57 >.776 α Null hypothesis is rejected. There were statistical 50 differences between groups. 5

26 7/09/ One-way ANOVA F While the t-test is limited to compare means of two groups, one-way ANOVA can compare more than two groups. Therefore, the t-test is considered a special case of one-way ANOVA One-way ANOVA F - Factor : Qualitative independent variable (number of values ; number of conditions). - Quantitative dependent variable (Interval or ratio variable). - Dependent variable normally distributed. - Independence of error effects. - Homogeneity of variance. - phericity (in longitudinal studies). 5 6

27 7/09/ One-way ANOVA F Types of hypothesis: General hypothesis : The purpose is to test for significant differences between group means, and this is done by analyzing the variances. It is a general decision. There is no specification of which pairs of groups are significantly different (concrete conditions). pecific hypothesis: There are specifications of which pairs of groups are significantly different (concrete conditions): Post hoc: firstly, the general hypothesis is tested; secondly, if null hypothesis is rejected, specific hypothesis are studied (section 3..3.). A priori: specific hypothesis are studied directly (section ). One-tailed vs. two-tailed hypothesis One-way ANOVA F Three steps to carry out: st. One-way ANOVA: (General hypothesis). ANOVA is an omnibustest statistic and cannot tell you which specific groups were significantly different from each other, only that at least two groups were. If null hypothesis is rejected: nd. Post-hoctest: (pecific hypothesis). To determine which specific groups differed from each other. 3 rd. TEP 3. tatistical power: goodness of fit. 54 7

28 7/09/ ANOVA table for one-way case ources of variation um of quares Degrees of Freedom Mean quare F Treatments Between groups K- b /df b Residuals/Error Within groups K (n-) W /df W M b /M w Total N Logic of ANOVA F crit (α 0.05 K-, K (n-)) 56 8

29 7/09/ One-way ANOVA: example Consider an experiment to study the effect of three different levels of some factor on a response (e.g. three types of fertilizer on plant growth). If we had 6 observations for each level, we could write the outcome of the experiment in a table like the following, where a, a, and a 3 are the three levels of the factor being studied. (α0.05) a a a One-way ANOVA example The null hypothesis, denoted H 0, for the overall F-test for this experiment would be that all three levels of the factor produce the same response, on average. To calculate the F- ratio: 58 9

30 7/09/ One-way ANOVA example a a a One-way ANOVA example tep : Calculate the mean within each group: 60 30

31 7/09/06 tep : Calculate the overall mean: One-way ANOVA example where ais the number of groups. tep 3: Calculate the "between-group" sum of squares: where nis the number of data values per group. The between-group degrees of freedom is the number of groups minus one df b 3 so the between-group mean square value is M B 84 / One-way ANOVA example tep 4: Calculate the "within-group" sum of squares. Begin by centering the data in each group a a a The within-group sum of squares is the sum of squares of all 8 values in this table W The within-group degrees of freedom is df W a(n ) 3(6 ) 5 Thus the within-group mean square value is 6 3

32 7/09/ One-way ANOVA example T df T 8-7 tep 5:The F-ratio is F crit (,5) 3.68 at α ince F 9.3 > 3.68, the results are significant at the 5% significance level. One would reject the null hypothesis, concluding that there is strong evidence that the expected values in the three groups differ One-way ANOVA example V df M F Between Within Total

33 7/09/ nd step. Post-hoc test: (pecific hypothesis): To determine which specific groups differed from each other. Post hoc tests can only be used when the 'omnibus' ANOVA found a significant effect. If the F-value for a factor turns out nonsignificant, you cannot go further with the analysis. This 'protects' the post hoc test from being (ab)used too liberally. They are designed to avoid hence of type I error. Tukey. cheffé. 65 DH Tukey: Post-hoc test Comparison of all pairs. Limitation: You can only compare pairs. HD test: Tukey's Honestly ignificant Difference. ψ ( HD) qα,wdf,k M n ( y i yj ) HDtukey There are sta s cally significant differences between groups. ( y i yj ) < HDtukey There are not sta s cally significant differences between groups. w Example: In the previous example, are there statistically significant differences between groups and 3? (α0.05) 66 33

34 7/09/ Post-hoc test: example Mw 4.5 ψ ( HD) qk ( n ), k q3(6 ),3 q5, * n 6 y y > 3.78 There are statistically significant differences between groups and cheffé: One-way ANOVA F Post-hoc tests More conservative. Recommended when few contrasts are developed. a j Y i Y i' ( k ) Fα, k,( n ) k Me n There are statistically significant differences between groups. a j Y i Y i' < ( k ) Fα, k,( n ) k Me n There are not statistically significant differences between groups. 68 p p 34

35 7/09/ One-way ANOVA F Post-hoc tests: example Example: Are there statistical differences between groups - and group 3? 69 ( k ) F F ( Y Y i, Y 0.05,,5 i' Y α, k,( n ) k One-way ANOVA F Post-hoc tests: example ) ( k ) F Y + Y M e α, k,( n ) k Y p n a * j 4.5 (3 ) F ,3,(6 )3 + + ( ) * >3 There are not statistical differences between groups - and group 3 M w p j n a 70 35

36 7/09/06 The p-value for this test is One-way ANOVA example After performing the F-test, it is common to carry out some "post-hoc" analysis of the group means. In this case, the first two group means differ by 4units, the first and third group means differ by 5units, and the second and third group means differ by only unit. Thus the first group is strongly different from the other groups, as the mean difference is more times the standard error, so we can be highly confident that the population of the first group differs from the population means of the other groups. However there is no evidence that the second and third groups have different population means from each other, as their mean difference of one unit is comparable to the standard error. Note F(x,y) denotes an F-distribution with xdegrees of freedom in the numerator and y degrees of freedom in the denominator A priori comparisons They are used when the general hypothesis is not of interest; only specific hypothesis are set out. nedecor F is not calculated. Types of a priori comparisons: Non-orthogonal contrasts: When lot of contrasts are done, type I error (α probability of being wrong when rejecting the null hypothesis) increases. When c>k-(cnumber of contrasts), correction of α is required. When c k-, some authors recommend the correction and others consider that it is not necessary. Orthogonal contrasts: α does not increase, so correction is not necessary (in post-hoc contrasts, the correction is not necessary either because cheffé and Tukey formulas already control this increase). 7 36

37 7/09/ A priori comparisons Formula to calculate bc : p n a j y j bc p a j A priori comparisons A. Orthogonal contrasts Characteristics: Their coefficients sum 0. If we multiply the coefficients of two orthogonal contrasts and sum the result, we also obtain 0. Tendency contrasts could be carried out: e.g., contrasts could be linear, quadratic and cubic. A F is going to be calculated for each contrast

38 7/09/ A priori comparisons A. Orthogonal contrasts k- possible contrasts. Example of coefficients to study tendency: a a a 3 a 4 Linear -3-3 Quadratic - - Cubic A priori comparisons A. Orthogonal contrasts Linear Cuadratic Cubic

39 7/09/ A priori comparisons A. Orthogonal contrasts Example : how to create orthogonal coefficients a a a 3 a 4 a 5 Contrast Contrast Contrast 3 Contrast A priori comparisons A. Orthogonal contrasts Example : how to create orthogonal coefficients a a a 3 a 4 a 5 Contrast Contrast Contrast Contrast

40 7/09/ A priori comparisons A. Orthogonal contrasts Example : how to create orthogonal coefficients a a a 3 a 4 a 5 Contrast Contrast Contrast 3 Contrast A priori comparisons A. Orthogonal contrasts Example : how to create orthogonal coefficients a a a 3 a 4 a 5 Contrast Contrast Contrast Contrast

41 7/09/ A priori comparisons A. Orthogonal contrasts Example(obtained from López, J., Trigo, M. E. y Arias, M. A.(999). Diseños experimentales. Planificación y análisis. evilla: Kronos): a a a Knowing that W 8, do this data present linear or quadratic tendency? (α0.05) A priori comparisons A. Orthogonal contrasts a a a y 78/ 6 3 y 48/ 6 8 y 4/

42 7/09/ A priori comparisons A. Orthogonal contrasts blinear n p p a a j j y j 6 [( )( 3) + ( 0)( 8) + ( + )( 4) ] ( ) + ( 0) + ( + ) bcuadratic n p p a a j j y j 6 [( + )( 3) + ( )( 8) + ( + )( 4) ] ( + ) + ( ) + ( + ) A priori comparisons A. Orthogonal contrasts V df M F Between 44 linear quadratic 0.7 Within Total 37 7 F (α, k-, k(n-)) F (0.05,,5) >4.54 There is linear tendency 0.7<4.54 There is not quadratic tendency 84 4

43 7/09/ A priori comparisons B. Non-orthogonal contrasts When lot of contrasts are done, type I error (α probability of being wrong when rejecting the null hypothesis) increases for the group of contrasts : Comparisons Type I error by contrast Type I error for the group of contrasts C C x 0. C x A priori comparisons B. Non-orthogonal contrasts Bonferroni: method for controlling the increase of α α α c C Being: α usually, 0.05 or 0.0 C number of contrasts Comparisons Type I error by contrast Type I error for the group of contrasts C 0.05/3 0.05/ C 0.05/3 (0.05/3) x 0.03 C3 0.05/3 (0.05/3) x The highest type I error for the group of contrasts is α 86 43

44 7/09/ One-way ANOVA F in longitudinal studies In a longitudinal study, each participant presents at least a score for each condition. The most important assumption to take into account is the sphericity. teps:. Testing assumptions.. One-way ANOVA F. 3. Effect size One-way ANOVA F in longitudinal studies tep. Testing assumptions: Normality:it is not important because F is robust even when it is violated. Independence of errors: it is usually violated because the measurements of the same person tends to be similar. Homoscedasticity: it is usually violated because the scores obtained closer in time are more similar. Although the two last assumptions are violated, F could be used if the relationship between variances and covariances are of sphericity

45 7/09/06 BETWEEN UBJECT WITHIN UBJECT tep. One-way ANOVA F V df M F n- n(k-) A k k- A /df A M A /M ( ) n y xa. j y.. ubjects x A (error) n k (n-)(k-) xa /df xa y + y y y TOTAL One-way ANOVA F in longitudinal studies n k i n i i i ( ) ij.. i.. j i j ( ) y i y k j k ( ) y ij y.. j... ( ) y y ij i. nk One-way ANOVA F in longitudinal studies tep. One-way ANOVA F Three stages test: It is used in order to try to avoid unnecessary calculations. Adjusted εhas to be calculated if we can not conclude based on the two first stages, and we have to carry out the third one. ε (epsilon measurement of the degree in which the assumption sphericity is violated) is going to be multiplied by the degrees of freedom of the theoretical F

46 7/09/ One-way ANOVA F in longitudinal studies tep. One-way ANOVA F tage : normal F (ε) (too easy to reject H 0 ). If H o is accepted, we do not have to do anything else. If H o is rejected, accepting this conclusion would imply to have a high level of possibility of committing type I error. We carry out stage. tage : conservative F; ε/(k-) (too difficult to reject H 0 ). If H o is rejected, we do not have to do anything else. If H o is accepted, we have to carry out stage 3. tage 3: F; /(k-)< ε < One-way ANOVA F in longitudinal studies Example. We would like to study the effect that different revisions of a material (a first revision; a second revision; a 3 third revision) produce in the memorization. Participants a a a

47 7/09/ One-way ANOVA F in longitudinal studies Participants a a a y. j 8 6 y i One-way ANOVA F in longitudinal studies y A [ ) + ( ) + (6 ) ] ( ) (8 3*6 9 (8 ) + (.67 ) + (.33 ) 3 + (.67 ) + (5 ) + (4.33 ) 3( )

48 7/09/ One-way ANOVA F in longitudinal studies Ax ( ) + ( ) + ( ) + ( ) + (+ 5 8) + ( ) + (8 + 8 ) + (+.67 ) + (0+.33 ) + (3+.67 ) + (6+ 5 ) + ( ) + (+ 8 6) + ( ) + ( ) + ( ) + (7+ 5 6) + ( ) One-way ANOVA F in longitudinal studies V df M F BETWEEN UBJECT WITHIN UBJECT 6.67 A ubjects x A (error) TOTAL

49 7/09/ One-way ANOVA F in longitudinal studies F (α, k-, (n-)(k-) F (0.05,, 0) 4. Three stages test: tage : normal F (ε). F (α, (k-)ε, (n-)(k-) ε F (0.05, *, 0*) >4. H o is rejected, so we have to carry out stage. tage : conservative F; ε/(k-)/(3-)0.5. F (α, (k-)ε, (n-)(k-) ε F (0.05, *0.5, 0*0.5) F (0.05,, 5) >6.6 H o is rejected, so we do not have to carry out stage 3. This is the final decision TEP 3. tatistical power After analyzing the existence of significant relation between variables, we have to verify whether our conclusionsare or not erroneousbecause of statistical power. The powerof a statistical test is the probability that the test will reject a false null hypothesis (i.e. that it will not make a type II error ). As power increases, the chances of a type II error decrease. The probability of a type II error is referred to as the false negative (β). Therefore, power is equal to β, which is equal to sensitivity

50 7/09/06 4. TEP 3. tatistical power DECIION H o is true Accept H o -α: Level of confidence H o is false β: Type II error Reject H o α : Type I error -β Power TEP 3. tatistical power Power analysis can be usedto calculate the minimum sample size required to accept the outcome of a statistical test with a particular level of confidence. It can also be used to calculate the minimum effect size that is likely to be detected in a study using a given sample size. In statistics, an effect size is a measure of the strength of the relationship between two variables in a statistical population, or a sample-based estimation of that quantity

51 7/09/06 4. TEP 3. tatistical power Pearson's correlation, often denoted r, is widely used as an effect size when two quantitative variables are available; for instance, if one was studying the relationship between birth weight and longevity. A related effect size is the coefficient of determination(r-squared R ). In the case of two variables, this is a measure of the proportion of variance shared by both, and varies from 0 to. An R of 0. means that % of the variance of a variable is shared with the other variable TEP 3. tatistical power Effect size: R B T Low: 0.8 Medium: [ ] High:

52 7/09/06 4. TEP 3. TATITICAL POWER. EFFECT IZE. GOODNE OF FIT ignificant effect Non-significant effect High effect size Low effect size The effect probably exists The statistical significance can be due to an excessive high statistical power The nonsignificance can be due to low statistical power The effect probably does not exist TEP 3. TATITICAL POWER. EFFECT IZE. GOODNE OF FIT: EXAMPLE Based on the example presented in slide 57 (first example of F), conclude about the statistical power. V df M F Between Within Total

53 7/09/06 4. TEP 3. TATITICAL POWER. EFFECT IZE. GOODNE OF FIT: EXAMPLE Medium effect size: R B T ignificant: 9.3 > 3.68 Conclusion: The effect probably exists 05 53