Grat MacEwa Uiversity STAT 5 Dr. Kare Buro Formula Sheet Descriptive Statistics Sample Mea: x = x i i= Sample Variace: s = i= (x i x) = Σ i=x i (Σ i= x i) Sample Stadard Deviatio: s = Sample Variace = s Estimatio Parameter Estimator SE(Estimator) Approximate Cofidece Iterval µ x σ s x ± t α/ σ µ µ x x + σ s ( x x ) ± t α/ + s µ µ x d σ d x d ± t α/ s d t-test Statistics Test Statistic for -sample t-test cocerig µ if σ is ukow t 0 = x µ 0 s df = Test Statistic for two sample t-test for Comparig Two Populatio Meas: t 0 = ( x x ) d 0, df = mi(, ) s + s Test Statistic for paired t-test for Comparig Two Populatio Meas: t 0 = x d d 0 ( sd ) df =
Oe-way Aalysis of Variace (-way ANOVA) Total Sum of Squares: T otalss = (x ij x) = x ij ( x ij ), df T otal = where x is the overall mea, from all k samples ad = + +... + k. Sum of Squares for treatmets: SST = i ( x i x), df T = k with x i = sample mea of observatios i sample i. Sum of Squares for Error: SSE = ( )s + ( )s +... + ( k )s k, with s i is the stadard deviatio of observatio from sample i. df E = k ANOVA Table Treatmets/Groups k SST MST = SST/(k ) MST/MSE Error k SSE MSE = SSE/( k) Total T otalss Aalysis of cotrasts i the ANOVA model: Correspodig estimator γ = C µ + C µ... + C k µ k, with C + C +... C k = 0 c = C x + C x... + C k x k, SE(c) = s p C + C +... + C k k with the pooled estimator for σ: s p = MSE. ( α)00% cofidece iterval for γ: c ± t SE(c), df = k Test statistic for a t-test cocerig γ: t 0 = c SE(c), df = k Multiple Comparisos (Boferroi Procedure) Experimet wise error rate =α, m = umber of comparisos, compariso wise error rate α = α/m, The the margi of error (ME) for comparig treatmet i with treatmet j is: ME ij = t df E α / MSE + i j
The Radomized Block Desig k treatmets, b blocks Total Sum of Squares: T otalss = (x ij x) = x ij ( x ij ), df T otal = where x is the overall sample mea. Sum of Squares for treatmets: SST = b( x T i x), df T = k with x T i = sample mea of observatios for treatmet i. Sum of Squares for blocks: SSB = k( x Bj x), df B = b with x Bj = sample mea of observatios for block j. Sum of Squares for Error: SSE = T otalss SST SSB, df E = k b + ANOVA Table for RBD Treatmets k SST MST = SST/(k ) MST/MSE Blocks b SSB MSB = SSB/(b ) MSB/MSE Error k b + SSE MSE = SSE/( k b + ) Total T otalss Factorial Desig (here: -way ANOVA) Factor A at a levels, factor B at b levels ANOVA Table for factorial desig Factor A a SS(A) MS(A) = SS(A)/(a ) MS(A)/MSE Factor B b SS(B) MS(B) = SS(B)/(b ) MS(B)/MSE Iter A B (a )(b ) SS(A B) MS(A B) = SS(A B)/(a )(b ) MS(A B)/MSE Error df E SSE MSE = SSE/df E Total T otalss df E = a b (a )(b ) + Test statistic for Model Utility Test: F = (SS(A) + SS(B) + SS(A B)) (a +b +(a )(b )) MSE, df = a + b + (a )(b ), df d = df E 3
No Parametric Methods Sig Test: Test Statistic S 0 = umber of measuremets supportig H a (for -tailed, the larger of the two). Wilcoxo Rak Sum Test Test statistic: Both sample sizes are at least 0, T =sum of raks for sample. Wilcoxo Siged Rak Test z 0 = T ( + +), ( + +) Test statistic: Sample size of pairs is at least 0, T + =sum of raks for positive differeces. Kruskal-Wallis H Test Test statistic: H 0 = R i = mea rak for sample i. ( + ) z 0 = T + (+) 4 (+)(+) 4 i ( R i + ), df = k, Multiple Compariso for Kruskal Wallis Test (Boferroi procedure) Choose experimet wise error rate α.. Whe comparig k distributios let m = k(k )/ ad the compariso wise error rate α = α/m.. Use the Wilcoxo Rak Sum test to compare every pair of treatmets at sigificace level α. The all sigificat results hold simultaeously at experimet wise error rate of α. Partial Correlatio: r yx x = r yx r x x r yx r x x r yx 4
Simple Liear Regressio SS xy = x i y i ( x i )( y i ), = x i ( x i ) ad SS yy = yi ( y i ) ( ) SSxy Equatio of least squares lie (regressio lie): slope: ˆβ =, itercept: ˆβ0 = ȳ ˆβ x ŷ = ˆβ 0 + ˆβ x Estimate for σ: SSE = SS yy ˆβ SS xy, df E =, s = Pearso s Correlatio Coefficiet: r = SS xy SSxx SS yy SSE Coefficiet of determiatio: R = SS yy SSE SS yy Test statistic for model utility test: t 0 = ˆβ s/, df = ( α) 00% CI for β : ˆβ ± t s, df E = SSxx A ( α)00% CI for E(y) at x = x p : ŷ ± t s A ( α)00% Predictio Iterval for y at x = x p : ŷ ± t s + (x p x), df E = + + (x p x), df E = Multiple Liear Regressio SSE = (y i ŷ i ), df E = k ˆσ = s = MSE = SSE k Adjusted coefficiet of determiatio: R a = k ( ) SSE SS yy Test statistic for model utility test: F 0 = (SS yy SSE)/k MSE df d = df E = k (from ANOVA table), df = k, ( α) 00% Cofidece Iterval for a slope parameter β i : ˆβi ± t s ˆβi, df = df E = k Test statistic for test about slope parameter β i : t 0 = ˆβ i s ˆβi, df = k 5