ࡱ> U A0bjbj Njj"/lhhh$ < < <PZ<<TYV?"D"DDDDZDXXX`bbbbbb$E ehXWNXXX_DDZD7___X DDhZD`_X`__dhԿZDJ? f~3 <Z$PԿ )0Yt`g]Կ_HANDY REFERENCE SHEET 2 HRP 259 Calculation Formulas for Sample Data: Univariate: Sample proportion:  Sample mean:  EMBED Equation.3  =  Sum of squares of x:  [to ease computation: EMBED Equation.3 ] Sample variance:  EMBED Equation.3 = =  Sample standard deviation:  EMBED Equation.3  ==  Standard error of the sample mean:  EMBED Equation.3 = 2. Bivariate Sum of squares of xy:  EMBED Equation.3  [to ease computation: EMBED Equation.3 ] Sample Covariance:  EMBED Equation.3 =  EMBED Equation.3  =  EMBED Equation.3  Sample Correlation:  EMBED Equation.3 = Hypothesis Testing The Steps: Define your hypotheses (null, alternative) Specify your null distribution Do an experiment Calculate the p-value of what you observed Reject or fail to reject (~accept) the null hypothesis The Errors  Power=1-( Confidence intervals (estimation) For a mean (2 unknown):  EMBED Equation.3  [if variance known or large sample size( EMBED Equation.3 ] For a paired difference (2 unknown):  EMBED Equation.3  [where  EMBED Equation.3  = the within-pair difference] For a difference in means, 2 independent samples (2 s unknown but roughly equal):  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  or  EMBED Equation.3  For a proportion:  EMBED Equation.3  For a difference in proportions, 2 independent samples:  EMBED Equation.3  For a correlation coefficient  EMBED Equation.3  For a regression coefficient:  EMBED Equation.3  [ EMBED Equation.3 ] Common values of t and Z Confidence level EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 90%1.811.731.701.681.661.6495%2.232.092.042.011.981.9699%3.172.852.752.682.632.58For an odds ratio: 95% confidence limits: EMBED Equation.3  For a risk ratio: 95% confidence limits: EMBED Equation.3   EMBED Equation.3  Corresponding hypothesis tests Test for Ho: = o (2 unknown):  EMBED Equation.3  Test for Ho: d = 0 (2 unknown):  EMBED Equation.3  Test for Ho: x- y = 0 (2 unknown, but roughly equal):  EMBED Equation.3  Test for Ho: p = po:  EMBED Equation.3  Test for Ho: p1- p2= 0:  EMBED Equation.3  Test for Ho: r = 0:  EMBED Equation.3  Test for: Ho:  = 0  EMBED Equation.3  Corresponding sample size/power Sample size required to test Ho: d = 0 (paired difference ttest):  EMBED Equation.3  Corresponding power for a given n:  EMBED Equation.3  Smaller group sample size required to test Ho: x  y = 0 (two sample ttest): (where r=ratio of larger group to smaller group)  EMBED Equation.3  Corresponding power for a given n:  EMBED Equation.3  Smaller group sample size required to test Ho: p1 p2 = 0 (difference in two proportions): (where r=ratio of larger group to smaller group)  EMBED Equation.3  Corresponding power for a given n:  EMBED Equation.3  Sample size required to test Ho: r = 0 (correlation/equivalent to simple linear regression): (where r=ratio of larger group to smaller group)  EMBED Equation.3  Corresponding power for a given n:  EMBED Equation.3  Common values of Zpower Zpower:.25.52.841.281.642.33Power:60%70%80%90%95%99%Linear regression Assumptions of Linear Regression Linear regression assumes that 1. The relationship between X and Y is linear 2. Y is distributed normally at each value of X 3. The variance of Y at every value of X is the same (homogeneity of variances) ANOVA TABLE Source of variation d.f. Sum of squares Mean Sum of Squares F-statistic p-valueBetween (k groups)k-1 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Go to Fk-1,nk-k chartWithinnk-k EMBED Equation.3  EMBED Equation.3 Total variationnk-1TSS= EMBED Equation.3  Coefficient of Determination:  EMBED Equation.3  = EMBED Equation.3  Source of variation d.f. Sum of squares Mean Sum of Squares F-statistic p-valueModel (k levels of X)k-1 EMBED Equation.3  EMBED Equation.3  EMBED Equation.3 Go to Fk-1,N-k chartErrorN-k EMBED Equation.3  EMBED Equation.3 Total variationN-1TSS= EMBED Equation.3 ANOVA TABLE FOR linear regression (more general) case Coefficient of Determination:  EMBED Equation.3   EMBED Equation.3  Probability distributions often used in statistics: T-distribution Given n independent observations EMBED Equation.3 ,  EMBED Equation.3   The Chi-Square Distribution  EMBED Equation.3 ; where Z~ Normal(0,1)  The F- Distribution  Fn,m= EMBED Equation.3  Summary of common statistical tests for epidemiology/clinical research: Choice of appropriate statistical test or measure of association for various types of data by study design. Types of variables to be analyzed  Statistical procedure or measure of associationPredictor (independent) variable/sOutcome (dependent) variable Cross-sectional/case-control studies BinaryContinuous T-test* CategoricalContinuous ANOVA* ContinuousContinuousSimple linear regressionMultivariate (categorical and continuous) ContinuousMultiple linear regressionCategoricalCategoricalChi-square testBinaryBinaryOdds ratio, Mantel-Haenszel ORMultivariate (categorical and continuous)BinaryLogistic regression Cohort Studies/Clinical Trials BinaryBinaryRelative risk CategoricalTime-to-eventKaplan-Meier curve/ log-rank testMultivariate (categorical and continuous)Time-to-eventCox-proportional hazards modelCategorical Continuousrepeated Repeated-measures ANOVAMultivariate (categorical and continuous)ContinuousrepeatedMixed models for repeated measures*Non-parametric tests are used when the outcome variable is clearly non-normal and sample size is small. Fishers exact test is used when the expected cells contain less than 5 subjects. Course coverage in the HRP statistics sequence: Choice of appropriate statistical test or measure of association for various types of data by study design. Types of variables to be analyzed  Statistical procedure or measure of associationPredictor (independent) variable/sOutcome (dependent) variable Cross-sectional/case-control studies BinaryContinuous T-test* CategoricalContinuous ANOVA* ContinuousContinuousSimple linear regressionMultivariate (categorical and continuous) Continuous Multiple linear regressionCategoricalCategoricalChi-square testBinaryBinaryOdds ratio, Mantel-Haenszel ORMultivariate (categorical and continuous)BinaryLogistic regression Cohort Studies/Clinical Trials BinaryBinaryRisk ratio CategoricalTime-to-eventKaplan-Meier curve/ log-rank testMultivariate (categorical and continuous)Time-to-eventCox-proportional hazards model (hazard ratios)Categorical Continuousrepeated Repeated-measures ANOVAMultivariate (categorical and continuous)ContinuousrepeatedMixed models for repeated measures*Non-parametric tests are used when the outcome variable is clearly non-normal and sample size is small. Fishers exact test is used when the expected cells contain less than 5 subjects. Corresponding SAS PROCs: Choice of appropriate statistical test or measure of association for various types of data by study design. Types of variables to be analyzed  Statistical procedure or measure of association SAS PROC Predictor Outcome Cross-sectional/case-control studiesBinaryContinuous T-test* PROC TTESTCategoricalContinuous ANOVA* PROC ANOVAContinuousContinuousSimple linear regressionPROC REGMultivariate (categorical /continuous) ContinuousMultiple linear regressionPROC GLM CategoricalCategoricalChi-square testPROC FREQBinaryBinaryOdds ratio, Mantel-Haenszel ORPROC FREQMultivariate (categorical/ continuous)BinaryLogistic regressionPROC LOGISTICCohort Studies/Clinical TrialsBinaryBinaryRisk ratioPROC FREQCategoricalTime-to-eventKaplan-Meier curve/ log-rank testPROC LIFETESTMultivariate (categorical and continuous)Time-to-eventCox-proportional hazards model (hazard ratios)PROC PHREGCategorical Continuousrepeated Repeated-measures ANOVAPROC GLMMultivariate (categorical and continuous)ContinuousrepeatedMixed models for repeated measuresPROC MIXED*Non-parametric equivalents: PROC NPAR1WAY; Fishers exact test: PROC FREQ, option: exact Handy Reference II PAGE  PAGE 18  INCLUDEPICTURE "http://www.bsu.edu/classes/leduc/ITMFG463/fcurve.jpg" \* MERGEFORMATINET  Type II Error (() E(n) = n Var(n) = 2n Variance rules for correlated random variables: Var (x+y)=Var(x)+Var(y)+2Cov(x,y); Var (x-y)=Var(x)+Var(y)-2Cov(x,y) Correct Do not reject H0 Correct Type I error (() Reject H0 H0 False H0 True True state of null hypothesis Your Statistical Decision HRP261 HRP262 HRP259 "Jijxy'()*,-/09:WXkl»¯ꋄ}ymjeC CJUVaJCJH* jNEHU j$EHU6OJQJ] j4 EHUjeC CJUVaJj CJEHUjeC CJUVaJ jCJUCJ j EHU6] jEHU jEHUjUeC CJUVaJ jU jEHU 6>*CJ]>*("#JVk:;uvB 4  !^ !^,-@0lmnopqstv)*+,-UVijklnoz jF$EHUjeC CJUVaJj!CJEHUjeC CJUVaJ jCJUCJCJ jEHUjueC CJUVaJOJQJ jEHU jEHUj+eC CJUVaJ jEHU jEHU6] jU jEHU-(*PR|vjjeC CJUVaJ H*OJQJ jb56OJQJ\]aJ05\jUmHnHu>*6] 6>*CJ] 5>*CJ\5 j.EHU jX+EHUjeC CJUVaJjCJUmHnHu jc(EHUjeC UV jU j8&EHUjeC CJUVaJ(;Lx&(  & Fh^h 4 & F$ 4RTV     L N h j  " $ J L N P V X |u j>EHUjeC CJUVaJ j;EHUjeC CJUVaJ 6>*]j9CJEHUjveC CJUVaJ j7EHUjeC CJUVaJ H*OJQJj4CJEHUjJeC CJUVaJ jCJU jCJCJ jU j2EHU.(  d f * , D E ] ^ | } `  !` !^X ~   E F Y Z [ \ } ~   ywrw 6>*]>* jUQEHUjLfC CJUVaJ jNEHUjeC CJUVaJ jKEHUj$G CJUVaJ jpHEHUjVeC CJUVaJ jEEHUjeC CJUVaJ jBEHUjeC UV jU j@EHUj eC CJUVaJ,   0 1 2 3 4 5 H I J K L M ` a b c d e x y z { | } üwp j`EHUj\{A CJUVaJ5\ j^EHUj]eC UV j\EHUjeC CJUVaJ jZEHUjeC CJUVaJ jXEHUjeC CJUVaJ jlVEHUjeC CJUVaJ jTTEHUjeC CJUVaJ jU 6>*],  4 L d | $$Ifa$ 5,,, $$Ifa$$$Ifl֞ D,"V\064 la $$Ifa$ 5,,, $$Ifa$$$Ifl֞ D,"V\064 la $$Ifa$ 5,,, $$Ifa$$$Ifl֞ D,"V\064 la  * $$Ifa$*,RT533-3`$$Ifl֞ D,"V\064 la 0246:<>dfhjvx&(*,żڴ} jlEHUjfC CJUVaJ jiEHUj|fC CJUVaJ H*OJQJ H*OJQJOJQJ5OJQJ^Jj_h5EHUjeC CJUVaJ5 j5U 5>*CJ\ jdEHUj\{A CJUVaJ 6CJ] jU18:<vlnh :<ln  !x` !^x`` *,@bhj  2468NPRX\`bnp "ü jxEHUj$G CJUVaJ jtEHUjzfC CJUVaJ jArEHUjfC CJUVaJCJ j4oEHUjPME CJUVaJ jU 6>*] H*OJQJ H*OJQJOJQJ;"h4d~+D&  !dh`  !dhdhdh` dhx`x``"$JLNPhj46\^`b246:>@HJ+,-@AvjE$G CJUVaJ jtEHUjD$G CJUVaJ jɀEHUjD$G CJUVaJ j#~EHUj.D$G CJUVaJEH j#EHU H*OJQJOJQJ5>*B*CJ\aJph>*B*aJph ja{EHUjfC CJUVaJ jU.ABCDopqsuvz{"#$%CDEG "#$)37Ľ~z5\ 6H*]6]H* 6>*H*] 6>*]>* j$EHUjvF$G CJUVaJ jyEHUjIF$G CJUVaJ jxEHUjE$G CJUVaJ j_EHUjE$G CJUVaJ H*OJQJOJQJEH j#EHU jvEHU0   #+/37<AF $$Ifa$  !dh`dh` dhx`FGNRV5,,, $$Ifa$$$Ifl֞ D,"V\064 la7VZyjk{2345=FMYZmǵ̌xi[̌56>*H*OJQJ\]j56>*EHU\]&jG$G 56>*CJUV\]aJj56>*U\]j>*EHUjG$G >*UV>* j>*UaJ jEHUjnG$G UV jU56>*\] 6>*]nHtH B*phB*PJaJph B*aJph>*B*aJph5\#VZ^bf $$Ifa$fgyz53111$$Ifl֞ D,"V\064 laklmnz{| $$Ifa$`^6<Foddd^ddVVx$If$If $x$Ifa$$$Iflֈ^  $[64 la FLMTYqg\\TT\\x$If $x$Ifa$$$Iflֈ^  $[64 la$Ifmnopqrȅ|pg|` jEHUj5EHUj\C CJUVaJ j5U5 6>*] j>EHUj G$G CJUVaJ jUj>*EHU&jG$G 56>*CJUV\]aJ>* j>*U56>*\]j56>*U\]jD56>*EHU\]&jreC 56>*CJUV\]aJ!odd^ddd$If $x$Ifa$$$Iflֈ^  $[64 la*+,1ommkk[[[[[$$X&@#$/Ifa$$$Iflֈ^  $[64 la noڕځrdڕP&jeC 56>*CJUV\]aJ56>*H*OJQJ\]jٯ56>*EHU\]&jeH$G 56>*CJUV\]aJj56>*U\]j>*EHU&jHH$G 56>*CJUV\]aJ>* j>*UaJ jEHUj H$G UV56>*\] 6>*] jU jEHUjeC CJUVaJ12ABVWXdefn$$X&@#$/Ifa$ novgUUUHUU $X&@#$/If$x$X&@#$/Ifa$$$IflֈM z"fS L 6@X64 laK99$x$X&@#$/Ifa$$$IflֈM z"fS L 6@X64 la $X&@#$/Ifx$X&@#$/If    !$%9=>QRSTUXYӿӜxo_jME >*CJEHUVaJj>*EHUj5>*EHUj\C >*CJUVaJj5>*U5>*5 jEHUjeC CJUVaJ jUj>*EHU&j[H$G 56>*CJUV\]aJ>* j>*U56>*\]j56>*U\]j756>*EHU\]! 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