Formulas and Tables for Gerstman

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1 Formulas ad Tables for Gerstma Measuremet ad Study Desig Biostatistics is more tha a compilatio of computatioal techiques! Measuremet scales: quatitative, ordial, categorical Iformatio quality is primary (GIGO) Data table: observatios, variables, values Use Table A or radom umber geerator to choose simple radom sample or radomize a treatmet Comparative studies must strive for group comparability to make valid ifereces. Comparative studies may be either experimetal or observatioal i desig: Beware lurkig variables, esp. i o-experimetal studies: cofoudig! Explorig ad Describig Data Explore distributioal shape, locatio, ad spread; check for outliers Frequecy, relative frequecy, cumulative frequecy Sample mea: x xi Media: Form a ordered array. The media is the value with a depth of + ; whe is odd, average the two middle values. Quartiles (Tukey s higes): Divide the ordered array at the media; whe is odd, the media belogs to both the low group ad the high group. Q is media of the low group. Q3 is the media of the high group. IQR Q3 Q Boxplot: plot media ad quartiles (box); determie feces: F L Q.5 IQR, F U Q3 +.5 IQR; plot outside values (if ay); draw whiskers from higes to iside values Five-poit summary: miimum, Q, media, Q3, maximum Sample stadard deviatio: s ( xi x) C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of

2 Probability Basic properties: () 0 Pr(A) ; () Pr(S) ; (3) Pr(Ā) Pr(A) (4) Pr(A or B) Pr(A) + Pr(B) for disjoit evets x Biomial: X b(, p), Pr( X x) C p q where µ p ad σ pq where q p x x C x! x!( x)! To fid probabilities o X N(μ, σ): () State () Stadardize z x μ (3) Sketch (4) Use Table B σ To fid percetile values o X N(μ, σ): () State () Sketch (3) Table B (4) Ustadardize: x μ + z p σ Samplig Distributios ad Itroductio to Iferece Samplig distributio of mea: x N(μ, σ ) whe populatio Normal or sample large Hypothesis testig procedure: (A.) H 0 ad H a (B.) Test statistic (C.) P-value (D.) Optioal: Sigificace level x μ0 σ Oe-sample test of mea (σ kow): H 0 : μ μ 0 ; z where SE x SE x σ Cofidece iterval for μ (σ kow): x ± z α / SE x where SE x Coditios for z procedures: SRS, Normal populatio or large sample, σ kow Iferece about Meas Sigle samples ad match-pairs (matched-pairs, aalyze the delta variable ) x μ0 s To test H 0 : μ μ 0 use tstat where SE x ad df SE ( α)00% Cofidece iterval for μ: α x x x ±, t SE Sample size ad power to limit margi of error m, use σ z α m σ Sample size to detect a differece Δ with stated power ad α, use Power of a test to detect Δ at give α: β Φ z Two idepedet samples α Δ + σ C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of ( z β + z α ) x x s s To test H 0 : μ μ use tstat where SE x SE x x x + df coservative smaller of ( ) or ( ) [use df Welch whe workig with a computer] ( α)00% CI for μ μ : ( x x) ± ( t df α )( SE x x ) σ z To estimate μ μ with margi of error m, use α i each group m, σ z β + z α Sample size test H 0 : μ μ at give β ad α: use i each group Δ ( ) Δ

3 If ot equal group sizes are ot possible, calculate, fix, ad use k idepedet samples MSB To test H 0 : μ μ μ k : F stat with df B ad df W via ANOVA table MSW Variace Sum of Squares df Mea Square k Betwee SS B SSB ( ) groups i xi x df B k MSB df Withi groups SS W i k i i ( ) s df W N k Total SS T SS B + SS W df df B + df W xi x j Post hoc least square differece; tstat where SE SE i x i x j Cofidece iterval for μ i μ j ( x x ) ± ( t df α )( SE x x ), xi x j Post hoc Boferroi: multiply P by umber of comparisos ad use MSW i t α N k, c Correlatio ad Regressio (use calculator or computer) Correlatio coefficiet r z X zy To test H 0 : ρ 0, use t stat Cofidece iterval for ρ: Regressio lie: y ˆ a + bx s Slope estimate: b r s Itercept estimate: a y bx Y X r r where SE r ad df SE r r ϖ r + ϖ LCL ad UCL where ϖ rϖ + rϖ B SS MSW df W W + ; df N k j i the CI formula t α df, t α df, Stadard error of the regressio s Y x residuals with df sy x ( α)00% cofidece iterval for β b ± (t -,-α/ )(SE b ) where SEb s b To test H 0 : β 0, use t stat with df SE b Multiple regressio model: y ˆ a + b x + bx + + b k xk (determie regressio coefficiets with computer program) X + df C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 3 of

4 Iferece about Proportios Sigle sample umber of successes Sample proportio: pˆ ( α)00% CI for p p ± z pq α where x + p ad q p + 4 pˆ p0 To test H 0 : p p 0 use (use exact biomial procedure i small samples) p q z stat 0 0 z Sample size requiremet to limit the margi of error (m): use * * p q α m Two idepedet samples Notatio (-by- cross-tabulatio) + Total Exposed a b No-exposed a b Total m m N Sample proportios (cohort ad aturalistic samples) a a p ˆ ad p ˆ To test H 0 : p p, (z stat ad X stat are equivalet for -by-) pˆ pˆ umber of successes, both samples combied z stat where p total observatios, both samples combied pq + ( O E ) i i X stat where all Ei row total colum total E i with df (R ) (C ) table total I small samples, use Fisher s exact test or the Mid-P modificatio (computer) Risk differece pˆ pˆ (do ot use i case-cotrol sample) ( α)00% CI p p ( p p) ± z α SE p p where Risk ratio ai + p i ad SE + i p q p p + p q ˆ pˆ a / R R (do ot use i case-cotrol sample) pˆ a / ( α)00% CI for RR e l RR ˆ ± z SE α l RR ˆ where SE l R ˆ R + a a C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 4 of

5 Odds ratio ˆ a OR a b b ( α)00% CI for the OR e lor ˆ ± z SE α l OR ˆ where SE l O ˆ R a b a b Matched-pairs Notatio (for case-cotrol data) Case E+ Case E Cotrol E+ a b Cotrol E c d c O ˆ R b ( α)00% CI for the OR e To test H 0 : OR, use lor ˆ ± z SE α l OR ˆ ( c b) z stat c + b where SE + c b l O ˆ R C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 5 of

6 Table A. Two thousad radom digits Lie C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 6 of

7 Table B: Cumulative probabilities for a Stadard Normal Z variable; traditioal z table. z C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 7 of

8 Table C. Traditioal t table; tables etries represet t values. Cumulative Upper tail df (z) % 60% 70% 80% 90% 95% 98% 99% 99.5% 99.8% 99.9% Cofidece level: ( α/)00% C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 8 of

9 Table E. Chi-square table Upper tail P df C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 9 of

10 Table F: Two tails of z; table etries are two-sided P-values for z. z C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page 0 of

11 Table G: Two-sided P-values from t statistics ("o-traditioal t table") df ( to 5) t df (6 to 30) t C:\data\biostat-text\Formulas_ad_Tables_for_Gerstma.doc Page of

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics

TABLES AND FORMULAS FOR MOORE Basic Practice of Statistics Explorig Data: Distributios Look for overall patter (shape, ceter, spread) ad deviatios (outliers). Mea (use a calculator): x = x 1 + x 2 + +