Nonparametric Nonparametric TestTest

Teaching Assistant: Teaching Assistant: Zuo Xiaoyu Zuo Xiaoyu

Chapter 7

OutlineOutline

Summary of the hypothesis Summary of the hypothesis testing methods learned so fartesting methods learned so far

Computer experiments of Computer experiments of different types of different types of Nonparametric test based on Nonparametric test based on rank.rank.

Interclass practicesInterclass practices

Questions for this chapterQuestions for this chapter

1.1. What is What is nonparametric test?nonparametric test?

2.2. What is theWhat is the difference difference between between

parametric test and nonparametric test?parametric test and nonparametric test?

3.3. WhenWhen should we use parametric test should we use parametric test

and when should we use nonparametric and when should we use nonparametric

test?test?

4.4. Different designsDifferent designs for nonparametric for nonparametric

test we have learned?test we have learned?

Summary of Summary of methodsmethodsFLOWCHARTFLOWCHART

Several points for Several points for the flowchartthe flowchart

1. Different types of data1. Different types of data

Cardinal data (continuous variable)Cardinal data (continuous variable)

Ordinal data (ordinal variable)Ordinal data (ordinal variable)

Nominal data (categorical variable)Nominal data (categorical variable)

2. Decision rule of 2. Decision rule of nonparametric testnonparametric test

Assumptions met?Assumptions met?

1.1. follows normal distribution , follows normal distribution , ; ;

2.2. homogeneity of variances, . homogeneity of variances, . LEVENE’S VARIANCE EQUALIT TEST

NORMALITY TEST OR Q-Q PLOT

2( , )i iN 1,2, , i G

2 2 21 2 G

ijX

For 3 or more samples

3. Comparison of parametric test and nonp3. Comparison of parametric test and nonparametric test under different design:arametric test under different design:

Mann-Whitney U Test(2 independent samples)Mann-Whitney U Test(2 independent samples) Wilcoxon Signed-Rank Test (2 related sampleWilcoxon Signed-Rank Test (2 related sample

s)s) Kruskal-Wallis Test (k independent samples)Kruskal-Wallis Test (k independent samples)

Two independent sample t-testPaired sample t-test

ANOVA

Wilcoxon Rank Sum TestWilcoxon Rank Sum Test

4. Multiple comparison 4. Multiple comparison problemproblem

For parametric test:For parametric test: LSD-LSD-tt test test SNK-SNK-q q testtest

Bonferroni Correction for Bonferroni Correction for αα

For nonparametric test based on rank:For nonparametric test based on rank:

Bonferroni Correction for Bonferroni Correction for αα

Computer experiment Computer experiment sectionsection

Three examples

Example 7.1Example 7.1 A senior registrar in the rheumatology A senior registrar in the rheumatology

clinic of a district hospital has designed a clinic of a district hospital has designed a clinical trial of a new drug for rheumatoid clinical trial of a new drug for rheumatoid arthritis. 20 patients were randomized into arthritis. 20 patients were randomized into two groups of ten to receive either the two groups of ten to receive either the standard therapy A or a new treatment B. The standard therapy A or a new treatment B. The plasma globulin fractions after treatment are plasma globulin fractions after treatment are listed in table 7.1 listed in table 7.1

Tabel 7.1 plasma globulin fraction after randomizTabel 7.1 plasma globulin fraction after randomiz

ation to treatments A or Bation to treatments A or B

We wish to test whether the new treatment has changed the plasma globulin, and we are worried about the assumption of Normality.

TreatmeTreatmentAntA

3388

2266

2299

4411

3366

3311

3322

3300

3535 3333

TreatmeTreatmentBntB

4455

2288

2277

3388

4400

4422

3399

3399

4040 4455

Design?

Steps of analysis?Data: EX-7.1.SAV

Two Independent Two Independent SamplesSamplesNormality met?

Equal variance met?

Data: EX-7.1.SAV

7.1.2 Data File7.1.2 Data File Variable Name: Variable Name: globulinglobulin; ; Variable Label: Plasma globulin fraction;Variable Label: Plasma globulin fraction; Variable Name: Variable Name: groupgroup Value Label(1:treatment A; 2:treatment B)Value Label(1:treatment A; 2:treatment B)

7.1.3 Procedure7.1.3 Procedure From the menus, choose: Analyze Non-From the menus, choose: Analyze Non-

parametric Test 2 independent samples, parametric Test 2 independent samples, open “Two-Independent-Sample Test” open “Two-Independent-Sample Test” dialog box.dialog box.

7.1.3 Procedure7.1.3 Procedure In “Two-Independent-Sample Test” dialog box:In “Two-Independent-Sample Test” dialog box:Click on variable Click on variable globulinglobulin to highlight it, move it to the to highlight it, move it to the

“Test variable List” box by clicking the right-pointing “Test variable List” box by clicking the right-pointing arrow between the boxes;arrow between the boxes;

Click on variable Click on variable groupgroup to highlight it, move it to the to highlight it, move it to the “Grouping Variable” box by clicking the right-pointing “Grouping Variable” box by clicking the right-pointing arrow between the boxes;arrow between the boxes;

Click on button, type “1” in the box of Group 1 Click on button, type “1” in the box of Group 1 and type “2” in the box of Group 2, click on and type “2” in the box of Group 2, click on button;button;

Click on button.Click on button.

7.1.4 Output and 7.1.4 Output and InterpretationInterpretation

The data is analyzed by the Mann-Whitney U The data is analyzed by the Mann-Whitney U test. The results indicate that there is a test. The results indicate that there is a statistically significant difference in plasma statistically significant difference in plasma globulin fraction between treatment A and globulin fraction between treatment A and treatment B treatment B (z=2.01, p=0.045).(z=2.01, p=0.045). You can You can determine which group has the higher rank by determine which group has the higher rank by looking at “Mean Rank’. The mean rank of looking at “Mean Rank’. The mean rank of treatment B is larger than that of treatment A.treatment B is larger than that of treatment A.

Triglycerides are blood constituents that are Triglycerides are blood constituents that are thought to play a role in coronary artery disease. thought to play a role in coronary artery disease. To see whether regular exercise could reduce To see whether regular exercise could reduce triglyceride levels, researchers measured the triglyceride levels, researchers measured the concentration of triglycerides in the blood serum concentration of triglycerides in the blood serum of 7 male volunteers, before and after of 7 male volunteers, before and after participation in a 10-week exercise program. The participation in a 10-week exercise program. The results are shown in table 7-2. note that results are shown in table 7-2. note that there is there is considerable variation from one participant to considerable variation from one participant to anotheranother. For instance, participant 1 had relatively . For instance, participant 1 had relatively low triglyceride levels both before and after, low triglyceride levels both before and after, while participant 3 had relatively high levels.while participant 3 had relatively high levels.

7.2 Example7.2 Example

Table 7-2 Serum Triglycerides (mmol/L)Table 7-2 Serum Triglycerides (mmol/L)

ParticipaParticipantnt

11 22 33 44 55 66 77

BeforeBefore 0.80.877

1.11.133

3.13.144

2.12.144

2.92.988

1.11.188

1.61.600

AfterAfter 0.50.577

1.01.033

1.41.477

1.41.433

1.21.200

1.01.099

1.51.511

Design?

Steps of analysis?Data: EX-7.2.SAV

Two Related Samples Two Related Samples (paired design)(paired design)

Normality met?

Equal variance needed?

Data: EX-7.2.SAV

7.2.2 Data File7.2.2 Data File

Variable Name: Variable Name: before; after.before; after.

7.2.3 Procedure7.2.3 Procedure From the menus, choose: Analyze From the menus, choose: Analyze

Non-parametric Test 2 related Non-parametric Test 2 related samples”, open “Two-Related-Sample samples”, open “Two-Related-Sample Tests” dialog box.Tests” dialog box.

7.2.3 Procedure7.2.3 Procedure In “Two-Related-Sample Tests” dialog box:In “Two-Related-Sample Tests” dialog box:Click on variable Click on variable beforebefore and and afterafter to highlight them, to highlight them,

move them together to the “Test Pairs List” box bmove them together to the “Test Pairs List” box by clicking the right-pointing arrow between the boy clicking the right-pointing arrow between the boxes;xes;

Choose “Wilcoxcon” in Test Type (default);Choose “Wilcoxcon” in Test Type (default);Click on button.Click on button.

7.2.4 Output and 7.2.4 Output and InterpretationInterpretation

The data is analyzed by the Wilcoxon Signed-Rank Test. The results indicate that there is a statistically significant difference in Serum Triglycerides before and after participation in a 10-week exercise program (z=2.37, p=0.018).

7.3 Example7.3 Example DNA content in gastric mucosal cells among 4 DNA content in gastric mucosal cells among 4

kinds of persons are shown in table 7-3. Is there a kinds of persons are shown in table 7-3. Is there a significant difference in DNA content in gastric mucosal significant difference in DNA content in gastric mucosal cells among 4 kinds of persons. cells among 4 kinds of persons.

Table 7-3 DNA content in gastric mucosal cells Table 7-3 DNA content in gastric mucosal cells among 4 kinds of personsamong 4 kinds of persons

KindKinds s

DNA content (A.U)DNA content (A.U)

HH 11.11.99

13.13.44

9.09.0 10.10.77

13.13.77

12.12.22

12.12.88

GMGMHH

13.13.99

17.17.22

16.16.55

14.14.77

14.14.66

13.13.00

12.12.00

16.16.44

14.14.11

EGCEGC 20.20.33

17.17.88

23.23.44

17.17.11

32.32.22

20.20.66

23.23.55

13.13.44

27.27.22

AGCAGC 25.25.11

28.28.66

27.27.22

22.22.99

19.19.99

23.23.99

23.23.11

21.21.11

15.15.66

19.19.44

18.18.88

16.16.44

k Independent k Independent samplessamples

Data: EX-7.3.SAV

Normality met?

Equal variance needed?

7.3.2 Data File7.3.2 Data File Variable Name: Variable Name: dnadna; Variable label (DNA content ; Variable label (DNA content

A.U)A.U) Variable Name: Variable Name: groupgroup; Value label (1:healthy; 2: ; Value label (1:healthy; 2:

hyperplasia; 3: early cancer 4: advanced cancer)hyperplasia; 3: early cancer 4: advanced cancer)

7.3.3 Procedure7.3.3 Procedure From the menus, choose: Analyze Non-parametric Test From the menus, choose: Analyze Non-parametric Test

k independent samples, open “Test for Several Independ k independent samples, open “Test for Several Independent Samples” dialog box.ent Samples” dialog box.

In “Test for Several Independent Samples” dialog box:In “Test for Several Independent Samples” dialog box:Click on variable Click on variable dna dna to highlight it, move it to the “Test Varito highlight it, move it to the “Test Vari

able List” box by clicking the right-pointing arrow between able List” box by clicking the right-pointing arrow between the boxes;the boxes;

Click on variable Click on variable groupgroup to highlight it, move it to the “Groupi to highlight it, move it to the “Grouping Variable” box by clicking the right-pointing arrow betwng Variable” box by clicking the right-pointing arrow between the boxes;een the boxes;

Click on button, type “1” in the box of Minimum aClick on button, type “1” in the box of Minimum and type “4” in the box of Maximum, click on button;nd type “4” in the box of Maximum, click on button;

Click on button.Click on button.

7.3.4 Output and 7.3.4 Output and InterpretationInterpretation

The data is analyzed by the Kruskal-Wallis Test. TThe data is analyzed by the Kruskal-Wallis Test. The results indicate that there was a statistically she results indicate that there was a statistically significant difference in DNA content in gastric ignificant difference in DNA content in gastric mucosal cells among 4 kinds of persons (chi-sqmucosal cells among 4 kinds of persons (chi-square=24.7, P<0.001).uare=24.7, P<0.001).

Exercises in classExercises in class

Exercise 1Exercise 1

14 newborn infants were grouped into 4 14 newborn infants were grouped into 4 categories according to their mother’s categories according to their mother’s smoking habit. smoking habit.

A: smoking more than 20 cigarettes per A: smoking more than 20 cigarettes per day; day;

B: smoking less than 20 cigarettes per day; B: smoking less than 20 cigarettes per day;

C: ex-smoker; C: ex-smoker;

D: never smoking. D: never smoking.

Their weights are listed in Table 12.7. Their weights are listed in Table 12.7.

Table 12.7 The weights of newborn infants grouped by their mothers’ smoking habit

Weight ijx A B C D 2.7 2.9 3.3 3.5 2.4 3.2 3.6 3.6 2.2 3.2 3.4 3.7 3.4 3.4

Table 12.4 Survival time of cats and rabbits without oxygen Cats Rabbits

Survival time Survival time 25 15 34 15 44 16 46 17 46 19

21 21 23 25 27 28 28 30 35

Exercise 2

Exercise 3Exercise 3

The investigator carry out the experiment The investigator carry out the experiment to compare the anti-tumor effects of three to compare the anti-tumor effects of three anti-tumor drugs A, B, C on mice sarcomaanti-tumor drugs A, B, C on mice sarcoma（肉瘤）（肉瘤） . 15 mice of the same race were s. 15 mice of the same race were selected and three anti-tumor drugs A, B, C elected and three anti-tumor drugs A, B, C randomly allocated into 3 mice within the randomly allocated into 3 mice within the same block.same block.

With the observations of sarcoma’s weight, the With the observations of sarcoma’s weight, the experiment results are shown in Table11.11. Pleaexperiment results are shown in Table11.11. Please se test if the effects of three anti-tumor drugs are test if the effects of three anti-tumor drugs are different.different.

Table 11.11 The weight of mice sarcoma with different drugs (g)

Drugs Block

A B C Total (Bi)

1 0.82 0.65 0.51 1.98 2 0.73 0.54 0.23 1.50 3 0.43 0.34 0.28 1.05 4 0.41 0.21 0.31 0.93 5 0.68 0.43 0.24 1.35

Total (Ti) 3.07 2.17 1.57 6.81 Sum of Squares (Qi) 2.02 1.06 0.55 3.63

AssignmentAssignment

P 204 N. 1P 204 N. 1

N. 3N. 3

Hand your homework on Monday afternoon (in class) or Tuesday noon (3th floor,

the building of Public Health School)

Thank you!!!Thank you!!!