#### Tea Tasting #### #### Fisher's Exact test a quick solution in R #### #### More details on this example corresponding to SAS output ### ########################################## #### data entry tea<- matrix(c(3, 1, 1, 3), ncol= 2, dimnames = list(Truth = c("Tea","Milk" ),Lady_says = c("Tea first","Milk first"))) #### one-sided Fisher's exact test fisher.test(tea, alternative = "greater") #### two-sided Fisher's exact test fisher.test(tea) #### OUTPUT for one-sided test ######### #### Fisher's Exact Test for Count Data #### #### data: tea #### p-value = 0.2429 #### alternative hypothesis: true odds ratio is greater than 1 #### 95 percent confidence interval: #### 0.3135693 Inf #### sample estimates: #### odds ratio #### 6.408309 ######## CONCLUSION ######## #### We cannot reject the null hypothesis, that is there is not enough evidence to establish association ###### What is the conclusion for two-sided test? ############## ################################################################ #### Another approach #### TeaLady=matrix(c(3, 1, 1, 3), ncol=2, dimnames=list(Poured=c("tea", "milk"), Lady=c("tea", "milk"))) TeaLady #### Pearson's Chi-squared test with Yates' continuity correction result=chisq.test(TeaLady) result #### Let's look at the observed, expected values and the residuals result\$observed result\$expected result\$residuals #### Let us look at the Percentage, Row Percentage and Column Percentage #### of the total observations contained in each cell. Contingency_Table=list(Frequency=TeaLady,Expected=result\$expected,Percentage=prop.table(TeaLady),RowPercentage=prop.table(TeaLady,1),ColPercentage=prop.table(TeaLady,2)) #### Pearson's Chi-squared test withOUT Yates' continuity correction result=chisq.test(TeaLady, correct=FALSE) result result\$observed result\$expected result\$residuals #### Likelihood Ratio Chi-Squared Statistic LR=2*sum(TeaLady*log(TeaLady/result\$expected)) LR #### p-value for the Likelihood Ratio Test LRchisq=1-pchisq(LR,df=1) LRchisq #### Fisher's Exact Test Fisher_Exact_TwoSided=fisher.test(TeaLady,alternative = "two.sided") Fisher_Exact_Less=fisher.test(TeaLady,alternative = "less") Fisher_Exact_Greater=fisher.test(TeaLady,alternative = "greater") list(Fisher_Exact_TwoSided=Fisher_Exact_TwoSided,Fisher_Exact_Less=Fisher_Exact_Less,Fisher_Exact_Greater=Fisher_Exact_Greater) RowSums=rowSums(TeaLady) ColSums=colSums(TeaLady) #### Column 1 Risk Estimates risk1_col1=TeaLady[1,1]/RowSums[1] risk2_col1=TeaLady[2,1]/RowSums[2] rho1=risk1_col1/risk2_col1 total1=ColSums[1]/sum(RowSums) diff1=risk1_col1-risk2_col1 list(Row1_Risk=risk1_col1,Row2_Risk=risk2_col1,Total=total1,difference=diff1) #### The confidence interval for the difference in proportions for column 1 SE_diff1=sqrt(risk1_col1*(1-risk1_col1)/RowSums[1]+risk2_col1*(1-risk2_col1)/RowSums[2]) CI_diff1=cbind(diff1-qnorm(0.975)*SE_diff1,diff1+qnorm(0.975)*SE_diff1) SE_diff1 CI_diff1 #### Column 2 Risk Estimates risk1_col2=TeaLady[1,2]/RowSums[1] risk2_col2=TeaLady[2,2]/RowSums[2] total2=ColSums[2]/sum(RowSums) diff2=risk1_col2-risk2_col2 list(Row1_Risk=risk1_col1,Row2_Risk=risk2_col1,Total=total1,difference=diff2) #### The confidence interval for the difference in proportions for column 2 SE_diff2=sqrt(risk1_col2*(1-risk1_col2)/RowSums[1]+risk2_col2*(1-risk2_col2)/RowSums[2]) CI_diff2=cbind(diff2-qnorm(0.975)*SE_diff2,diff2+qnorm(0.975)*SE_diff2) SE_diff2 CI_diff2 #### Estimate of the Odds of the two rows odds1=(TeaLady[2,1]/RowSums[2])/(TeaLady[1,1]/RowSums[1]) odds2=(TeaLady[2,2]/RowSums[2])/(TeaLady[1,2]/RowSums[1]) odds1 odds2 #### Odds Ratio oddsratio=odds2/odds1 oddsratio #### Confidence Interval of the odds ratio log_CI=cbind(log(oddsratio)-qnorm(0.975)*sqrt(sum(1/TeaLady)),log(oddsratio)+qnorm(0.975)*sqrt(sum(1/TeaLady))) CI_oddsratio=exp(log_CI) CI_oddsratio