One group sample - one sample t-test Test for value of a single mean E.g., test to see if mean SBP of all AIDHC employees is 120 mm Hg Assumptions –Parent population is normal –Sample observations (subjects) are independent
One group sample- one sample t- test Formula Let x 1, x 2, ….x n be a random sample from a normal population with mean µ and variance σ 2, then the following statistic is distributed as Student’s t with (n-1) degrees of freedom.
One group sample- one sample t- test Computation in Excel: –Excel does not have a 1-sample test, but we can fool it. –Suppose we want to test if the mean height of pediatric patients in our data set 1 is 50 inch –Create a dummy column parallel to the hgt column with an equal number of cells, all set to 0.0 –Run the Matched sample test using hgt and the dummy column and 50 as the hypothesized mean difference. –The p-value for two tail test is 0.0092
One group sample - one sample t-test Using SPSS: –Analyze> Compare Means >One Sample T Test > Select variable (e.g. height) > Test value: (e.g. 50) > ok –P-value is.009 –Interpretation: The mean height of the pediatric patients in our dataset 1 is statistically significantly different from 50 inches.
One group sample - Sign Test (Nonparametric) Use: (1) Compares the median of a single group with a specified value (instead of single sample t-test). Hypothesis: H 0 :Median = c H a :Median c H a :Median c Test Statistic: We take the difference of observations from median (x i - c). The number of positive difference follows a Binomial distribution. For large sample size, this distribution follows normal distribution.
One group sample - Sign Test (Nonparametric) SPSS: Analyze> Nonparametric Tests> Binomial
Two-group (independent) samples - two- sample t-statistic Use –Test for equality of two means Assumptions –Parent population is normal –Sample observations (subjects) are independent.
Two-group (independent) samples - two- sample t-statistic Formula (two groups) Formula (two groups) –Case 1: Equal Population Standard Deviations: The following statistic is distributed as t distribution with (n1+n2 -2) d.f. The pooled standard deviation, n1 and n2 are the sample sizes and S 1 and S 2 are the sample standard deviations of two groups.
Two-group (independent) samples - two- sample t-statistic Formula (two groups) –Case 2: Unequal population standard deviations The following statistic follows t distribution. The d.f. of this statistic is,
Two-group (independent) samples - two- sample t-statistic MS Excel (in Tools -> Data Analysis…) Two Groups (Independent Samples): –t-Test: Two-Sample Assuming Equal Variances –t-Test: Two-Sample Assuming Unequal Variances
Two-group (independent) samples - two- sample t-statistic Using SPSS: –Analyze>Compare Means>Independent-Samples T-test> –Select hgt as a Test Variable –Select sex as a Grouping Variable –In Define Groups, type f for Group 1 and m for Group 2 –Click Continue then OK –It gives us the p-value 0.205. We can assume equal variance as the p-value of F statistic for testing equality of variances is 0.845.
Two-group (independent) samples- Wilcoxon Rank-Sum Test (Nonparametric) Use: Compares medians of two independent groups. Corresponds to t-Test for 2 Independent Means Test Statistic: Let, X and Y be two samples of sizes m and n. Suppose N=m+n. Compute the rank of all N observations. Then, the statistic, W m = Sum of the ranks of all observations of variable X.
Two-group (independent) samples- Wilcoxon Rank-Sum Test (Nonparametric) Asthmatic score A Asthmatic score B ScoreRankScoreRank 711855 823 3.5824 3.5 772948 927979 886... Rank Sum19.525.5
Two-group (matched) samples - paired t- statistic Use: Compares equality of means of two matched or paired samples (e.g. pretest versus posttest) Assumptions: – Parent population is normal –Sample observations (subjects) are independent
Two-group (matched) samples - paired t- statistic Formula –The following statistic follows t distribution with n-1 d.f. Where, d is the difference of two matched samples and S d is the standard deviation of the variable d.
More on test statistic One-sided –There can only be on direction of effect –The investigator is only interested in one direction of effect. –Greater power to detect difference in expected direction Two-sided –Difference could go in either direction –More conservative
More on test statistic One group Two groups One sided A single mean differs from a known value in a specific direction. e.g. mean > 0 or median > 0 Two means differ from one another in a specific direction. e.g., mean 2 < mean 1 median 2 < median 1 Two sided A single mean differs from a known value in either direction. e.g., mean ≠ 0 or median 0 Two means are not equal. That is, mean 1 ≠ mean 2 median 1 ≠ median 2 median 1 ≠ median 2
Two-group (matched) samples Wilcoxon Signed-Rank Test (Nonparametric) USE : –Compares medians of two paired samples. Test Statistic –Obtain Difference Scores, D i = X 1i - X 2i –Take Absolute Value of Differences, D i –Assign Ranks to absolute values (lower to higher), R i –Sum up ranks for positive differences (T + ) and negative differences (T - ) Test Statistic is smaller of T - or T + (2-tailed)
Subject Hours of Sleep Difference Rank Ignoring Sign DrugPlacebo 188.8.131.52.5 27.07.9-0.93.5 184.108.40.2060 220.127.116.11 18.104.22.168 68.45.43.08 22.214.171.124 126.96.36.199 188.8.131.52 105.86.3-0.51 3 rd & 4 th ranks are tied hence averaged. P-value of this test is 0.02. Hence the test is significant at any level more than 2%, indicating the drug is more effective than placebo. Example of Wilcoxon signed rank test (two matched samples)
Two-group (matched) samples Wilcoxon Signed-Rank Test (Nonparametric) SPSS: –Two Matched Groups: Analyze> Nonparametric Tests> 2 Related Samples
Comparing > 2 independent samples: F statistic (Parametric) Comparing > 2 independent samples: F statistic (Parametric) Use: –Compares means of more than two groups –Testing the equality of population variances.
Comparing > 2 independent samples: F statistic (Parametric) Comparing > 2 independent samples: F statistic (Parametric) Let X and Y be two independent Chi-square variables with n 1 and n 2 d.f. respectively, then the following statistic follows a F distribution with n 1 and n 2 d.f. Let, X and Y are two independent normal variables with sample sizes n 1 and n 2. Then the following statistic follows a F distribution with n 1 and n 2 d.f. Where, s x 2 and s y 2 are sample variances of X and Y.
Comparing > 2 independent samples: F statistic (Parametric) Comparing > 2 independent samples: F statistic (Parametric) Hypotheses: Hypotheses: H 0 : µ 1 = µ 2 =…. =µ n H a : µ 1 ≠ µ 2 ≠ …. ≠µ n Comparison will be done using analysis of variance (ANOVA) technique. ANOVA uses F statistic for this comparison. The ANOVA technique will be covered in another class session.
Proportion Tests Use –Test for equality of two Proportions E.g. proportions of subjects in two treatment groups who benefited from treatment. –Test for the value of a single proportion E.g., to test if the proportion of smokers in a population is some specified value (less than 1)
Proportion Tests Formula –One Group: –Two Groups:
Proportion of males in Dataset 1 SPSS: –recode sex as numeric - Transform> Recode>Into Different Variables> Make all selections there and click on Change after recoding character variable into numeric. –Analyze> Nonparametric test> Binomial> select Test variable> Test proportion Set null hypothesis = 0.5 The p-value = 1.0
Chi-square statistic USE –Testing the population variance σ 2 = σ 0 2. –Testing the goodness of fit. –Testing the independence/ association of attributes Assumptions –Sample observations should be independent. –Cell frequencies should be >= 5. –Total observed and expected frequencies are equal
Chi-square statistic Formula: If x i (i=1,2,…n) are independent and normally distributed with mean µ and standard deviation σ, then, If we don’t know µ, then we estimate it using a sample mean and then,
Chi-square statistic For a contingency table we use the following chi- square test statistic,
Chi-square statistic MaleO(E)FemaleO(E)Total Group 1 9 (10) 20 Group 2 8 (10) 12 (10) 20 Group 3 11 (10) 9(10)20 303060
Chi-square statistic – calculation of expected frequency To obtain the expected frequency for any cell, use: Corresponding row total X column total / grand total E.g: cell for group 1 and female, substituting: 30 X 20 / 60 = 10
Chi-square statistic SPSS: –Analyze> Descriptive stat> Crosstabs> statistics> Chi-square –Select variables. –Click on Cell button to select items you want in cells, rows, and columns.
Credits Thanks are due to all whose works have been consulted prior to the preparation of these slides.