Nonparametric testing:Data is nonparametric if one of the following applies:the statistic can be used on categorical
scale data,the statistic can be used on ordinal scale data,the statistic can be used on a random variable of unspecified
distribution(ex.ordinal)
X2 Chi Square test:Univariate test that compares expected to observed frequencies of successes for more than two
possible outcomes.Always one-tailed.[Only IF EXPECT P to EQUAL for all(2) cases]Null is based on assumption of
( 𝑶−𝑬)𝟐
probability.Test Statistic ∑ (O is observed value, E is expected value).
𝑬
[0-E]=0,[O-E] ,[O-E] /E =X Decision rule=k-1(own table)Determined expected frequencies:Expected=Row
2 2 2
total*Column total/table total.
𝑛(𝑛+1)
Ordinal data: Rank from lowest to highest.Sum should always equal
2
Mann-Whitney Test:Tests if populations(medians)are equal or not.(Nonparametric equivalent of the 2-sample
test)Assumptions:ordinal or non-normal contiuous data,independent samples
Test Staitistic:U1= U 2=
Decision:n1=5,n2 =5,U=2 *Two-tailed table
EX.Hypothesis:H0 Income men<women; H1 Income men>Income women 2.Test Statistic 3.Decision Rule:Critical
𝑛(𝑛+1)
value;n1 =8,n2=10,U=20 4.Rank Values 5.Compute Test STATISTIC [CHECK ANSWER= ] Pick the lowest value for”U”
2
Wilcoxon signed-ranks test:Nonparametric equivalent of the paired t-test.Assumes ordinal or continuous
data(normal).Test Statistic is “T”the sum of differences of ranks with the least frequent sign.1.H0 :median difference=0
H1:median difference≠0 2.Decision:n=15,T=25(two-tail table)3.Subtract the two samples=Difference.Then rank
difference(+and-)Pick the lowest sum of the total ranks(2).
Compare more than 2 groups:Probability of type 1 error for at least 1 hypothesis is:1 – (1 – ) C =1-(-0.05) 3=0.14
Test with more than two independent samples, continuous variables, e.g., clinical trial with more than two comparison
groups (placebo, standard drug, experimental drug). This is an extension of the two-sample t-test.
Assumptions of ANOVA:Continuous outcome, normal distribution, or samples large enough to approximate
normality.Equality of variances in the k comparison groups
1.Set up hypothesis:H0:u1=u2=u3…..uk /H1 :means are not equal. Note Only one tailed
n j ( X j − X ) 2 /( k − 1)
F =
( X − X j ) 2 /( N − k)
3.Test statistic is:
Where nj is the sample size of the j th group (e.g., j = 1, 2, 3, and 4 with four groups),__is the sample mean of the j th
group, and__is the overall mean. Test statistic is computed by taking the ratio of what is called the “between -treatment”
variability to the “residual or error” variability.
4.Decision rule: df1=k-1,df2=N-k
5. Between-treatment sum of square:SSB= ∑ 𝑛(𝑋 − 𝑋) 2Error sums of squares:SSE=∑∑(X-XJ) 2Total sums of
squares:SST=∑∑(X-X) 2 =SSB+SSE
𝑺𝑺𝑩 𝑺𝑺𝑬 𝑴𝑺𝑩
Mean Squares:MSB= MSE= F Statistic:F=
𝒌−𝟏 𝑵−𝒌 𝑴𝑺𝑬
Sum of Squares(SS) Df Mean square(MS) F
Between treatments Sum divided by df=MS MS/MS=F
Error
Total
Kruskal-Willis Test:Non-parametric test of k(3+)comparison groups.Ordinal(ONLY KW) or continuous data.
𝟏𝟐
Test Statistic is:H=( ∑ R) − 3(𝑁 + 1).Where N is the total sample size,Rj and nj are the ranks and sample sizes of
𝑵(𝑵+𝟏) n
each of the k groups. 1. H 0 :the k population medians are equal H 1:the k population medians are not equal
2.Decision:K=3,nj of 8,5,2=5.805(PICK HIGHEST#)(have own table)(Higher than 8,USE CHI SQUARE TABLE=DF:K-
1)3.RANK, then total for each group 3.H=Test Statistic 4.Conc:H value is less than CV,Null accepted
scale data,the statistic can be used on ordinal scale data,the statistic can be used on a random variable of unspecified
distribution(ex.ordinal)
X2 Chi Square test:Univariate test that compares expected to observed frequencies of successes for more than two
possible outcomes.Always one-tailed.[Only IF EXPECT P to EQUAL for all(2) cases]Null is based on assumption of
( 𝑶−𝑬)𝟐
probability.Test Statistic ∑ (O is observed value, E is expected value).
𝑬
[0-E]=0,[O-E] ,[O-E] /E =X Decision rule=k-1(own table)Determined expected frequencies:Expected=Row
2 2 2
total*Column total/table total.
𝑛(𝑛+1)
Ordinal data: Rank from lowest to highest.Sum should always equal
2
Mann-Whitney Test:Tests if populations(medians)are equal or not.(Nonparametric equivalent of the 2-sample
test)Assumptions:ordinal or non-normal contiuous data,independent samples
Test Staitistic:U1= U 2=
Decision:n1=5,n2 =5,U=2 *Two-tailed table
EX.Hypothesis:H0 Income men<women; H1 Income men>Income women 2.Test Statistic 3.Decision Rule:Critical
𝑛(𝑛+1)
value;n1 =8,n2=10,U=20 4.Rank Values 5.Compute Test STATISTIC [CHECK ANSWER= ] Pick the lowest value for”U”
2
Wilcoxon signed-ranks test:Nonparametric equivalent of the paired t-test.Assumes ordinal or continuous
data(normal).Test Statistic is “T”the sum of differences of ranks with the least frequent sign.1.H0 :median difference=0
H1:median difference≠0 2.Decision:n=15,T=25(two-tail table)3.Subtract the two samples=Difference.Then rank
difference(+and-)Pick the lowest sum of the total ranks(2).
Compare more than 2 groups:Probability of type 1 error for at least 1 hypothesis is:1 – (1 – ) C =1-(-0.05) 3=0.14
Test with more than two independent samples, continuous variables, e.g., clinical trial with more than two comparison
groups (placebo, standard drug, experimental drug). This is an extension of the two-sample t-test.
Assumptions of ANOVA:Continuous outcome, normal distribution, or samples large enough to approximate
normality.Equality of variances in the k comparison groups
1.Set up hypothesis:H0:u1=u2=u3…..uk /H1 :means are not equal. Note Only one tailed
n j ( X j − X ) 2 /( k − 1)
F =
( X − X j ) 2 /( N − k)
3.Test statistic is:
Where nj is the sample size of the j th group (e.g., j = 1, 2, 3, and 4 with four groups),__is the sample mean of the j th
group, and__is the overall mean. Test statistic is computed by taking the ratio of what is called the “between -treatment”
variability to the “residual or error” variability.
4.Decision rule: df1=k-1,df2=N-k
5. Between-treatment sum of square:SSB= ∑ 𝑛(𝑋 − 𝑋) 2Error sums of squares:SSE=∑∑(X-XJ) 2Total sums of
squares:SST=∑∑(X-X) 2 =SSB+SSE
𝑺𝑺𝑩 𝑺𝑺𝑬 𝑴𝑺𝑩
Mean Squares:MSB= MSE= F Statistic:F=
𝒌−𝟏 𝑵−𝒌 𝑴𝑺𝑬
Sum of Squares(SS) Df Mean square(MS) F
Between treatments Sum divided by df=MS MS/MS=F
Error
Total
Kruskal-Willis Test:Non-parametric test of k(3+)comparison groups.Ordinal(ONLY KW) or continuous data.
𝟏𝟐
Test Statistic is:H=( ∑ R) − 3(𝑁 + 1).Where N is the total sample size,Rj and nj are the ranks and sample sizes of
𝑵(𝑵+𝟏) n
each of the k groups. 1. H 0 :the k population medians are equal H 1:the k population medians are not equal
2.Decision:K=3,nj of 8,5,2=5.805(PICK HIGHEST#)(have own table)(Higher than 8,USE CHI SQUARE TABLE=DF:K-
1)3.RANK, then total for each group 3.H=Test Statistic 4.Conc:H value is less than CV,Null accepted
