4.1. Lecture 4: Fitting distributions: goodness of fit. Goodness of fit: the underlying principle

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1 Lecture 4: Fttng dstrbutons: goodness of ft Goodness of ft Testng goodness of ft Testng normalty An mportant note on testng normalty! L4.1 Goodness of ft measures the extent to whch some emprcal dstrbuton fts the dstrbuton expected under the null hypothess Expected Fork length L4.2 Goodness of ft: the underlyng prncple Expected If the match between observed and expected s poorer than would be expected on the bass of measurement precson, then we should reject the null hypothess. Accept H 0 Reject H Fork length L

2 Testng goodness of ft : the Ch- square statstc (Χ( 2 ) Used for frequency data,.e. the number of observatons/results n each of n categores compared to the number expected under the null hypothess. 2 Χ = = 1 n ( f f ) f 2 Expected Category/class L4.4 How to translate Χ 2 nto p? Compare to the χ 2 dstrbuton wth n - 1 degrees of freedom. If p s less than the desred α level, reject the null hypothess. Probablty 0.3 χ 2 = 8.5, p = 0.31 accept p = α = χ 2 (df = 5) L4.5 Testng goodness of ft: the log lkelhood-rato Ch-square statstc (G)( Smlar to Χ 2, and usually gves smlar results. In some cases, G s more conservatve (.e. wll gve hgher p values). n f G = 2 f ln = 1 f Expected Category/class L

3 χ 2 versus the dstrbuton of Χ 2 or G For both Χ 2 and G, p values are calculated assumng a χ 2 dstrbuton......but as n decreases, both devate more and more from χ 2. Probablty χ 2 /Χ 2 /G (df = 5) Χ 2 /G, very small n Χ 2 /G, small n χ 2 L4.7 Assumptons (Χ( 2 and G) n s larger than 30. Expected frequences are all larger than 5. Test s qute robust except when there are only 2 categores (df = 1). For 2 categores, both X 2 and G overestmate χ 2, leadng to rejecton of null hypothess wth probablty greater than α,.e. the test s lberal. L4.8 What f n s too small, there are only 2 categores, etc.? Collect more data, thereby ncreasng n. If n > 2, combne categores. Use a correcton factor. Use another test. Age (yrs) Expected More data Age (yrs) Expected Classes combned Expected L

4 For 2 categores, both X 2 and G overestmate χ 2, leadng to rejecton of null hypothess wth probablty greater than α (.e. test s lberal). Contnuty correcton: add 0.5 to observed frequences. Wllams correcton: dvde test statstc (G or Χ 2 ) by: 2 k 1 q = 1+ 6n( k 1) Correctons for 2 categores Age (yrs) Expected 20 5 Age (yrs) Expected L4.10 Used when there are 2 categores. No assumptons Calculate exact probablty of obtanng N - k ndvduals n category 1 and k ndvduals n category 2, wth k = 0, 1, 2,... N. The bnomal test Probablty Number of observatons Bnomnal dstrbuton, p = 0.5, N = 10 L4.11 An example: sex rato of beavers H 0 : sex-rato s 1:1, so p = 0.5 = q p(0 males, females) = p(1 male/female, 9 male/female) =.0195 p(9 or more ndvduals of same sex) =.0215, or 2.15%. Sample Males Females 9 1 Expected 5 5 therefore, reject H 0 L

5 Multnomal test Smple extenson of bnomal test for more than 2 categores Must specfy 2 probabltes, p and q, for null hypothess, p + q + r = 1.0. No assumptons......but so tedous that n practce Χ 2 s used. L4.13 Multnomal test: segregaton ratos Hypothess: both parents Aa, therefore segregaton rato s 1 AA: 2 Aa: 1 aa. So under H 0, p =.25, q =.50, r =.25 For N = 60, p <.001 Therefore, reject H 0. Genotype O E AA Aa aa 0 15 L4.14 Goodness of ft: testng normalty Snce normalty s an assumpton of all parametrc statstcal tests, testng for normalty s often requred. Tests for normalty nclude Χ 2 or G, Kolmogorov-Smrnov, Wlks-Shapro & Lllefors. Expected under hypothess of normal dstrbuton Category/class L

6 F % 2.28% Cumulatve dstrbutons Cumulatve normal densty functon Normal probablty densty functon Areas under the normal probablty densty functon and the cumulatve normal dstrbuton functon 68.27% 0 3σ 2σ σ µ σ 2σ 3σ L4.16 Χ 2 or G test for normalty Put data n classes (hstogram) and compute expected frequences based on dscrete normal dstrbuton. Calculate Χ 2. Requres large samples (k mn = 10) and s not powerful because of loss of nformaton. Expected under hypothess of normal dstrbuton Category/class L4.17 Non-statstcal assessments of normalty Do normal probablty plot of normal equvalent devates (NEDs) versus X. If lne appears more or less straght, then data are approxmately normally dstrbuted. NEDs X Normal Non-normal L

7 Komolgorov-Smrnov goodness of ft Compares observed cumulatve dstrbuton to expected cumulatve dstrbuton under the null hypothess. p s based on D max, absolute dfference, between observed and expected cumulatve relatve frequences. Cumulatve frequency D max X L4.19 An example: wng length n fles 10 fles wth wng lengths: 4, 4.5, 4.9, 5.0, 5.1, 5.3, 5.5, 5.6, 5.7, 5.8, 5.9, 6.0 cumulatve relatve frequences:.1,.2,.3,.4,.5,.6,.7,.8,.9, 1.0 Cumulatve frequency D max Wng length L4.20 Lllefors test KS test s conservatve for tests n whch the expected dstrbuton s based on sample statstcs. Llefors corrects for ths to produce a more relable test. Should be used when null hypothess s ntrnsc versus extrnsc. L

8 An mportant note on testng normalty! When N s small, most tests have low power. Hence, very large devatons are requred n order to reject the null. When N s large, power s hgh. Hence, very small devatons from normalty wll be suffcent to reject the null. So, exercse common sense! L