Chapter 17:Nonparametric

Statistics

LO1 Use both the small-sample and large-sample runs tests to determine whether the order of observations in a sample is random.

LO2 Use both the small-sample and large-sample cases of the Mann-Whitney U test to determine if there is a difference in two independent populations.

LO3 Use both the small-sample and large-sample cases of the Wilcoxon matched-pairs signed rank test to compare the difference in two related samples.

continued...

Learning Objectives

LO4 Use the Kruskal-Wallis test to determine whether samples come from the same or different populations.

LO5 Use the Friedman test to determine whether different treatment levels come from the same population when a blocking variable is available.

LO6 Use Spearman’s rank correlation to analyze the degree of association of two variables.

Learning Objectives

LO1

• The appropriateness of the data analysis depends on the level of measurement of the data gathered: nominal, ordinal, interval, or ratio

• Parametric Statistics are statistical techniques based on assumptions about the population from which the sample data are collected.– A fundamental assumption is that data being analyzed are randomly

selected from a normally distributed population. – It requires quantitative measurement that yield interval or ratio level

data.

Parametric vs. Nonparametric Statistics

LO1

• Nonparametric Statistics are based on fewer assumptions about the population and the parameters than are parameter statistics. – Because of this property nonparametric statistics are sometimes

called “distribution-free” statistics.– A variety of nonparametric statistics are available for use with

nominal or ordinal data.

Parametric vs. Nonparametric Statistics

LO1

• Sometimes there is no parametric alternative to the use of nonparametric statistics.

• Certain nonparametric test can be used to analyze nominal data.

• Certain nonparametric test can be used to analyze ordinal data.

• The computations on nonparametric statistics are usually less complicated than those for parametric statistics, particularly for small samples.

• Probability statements obtained from most nonparametric tests are exact probabilities.

Advantages of Nonparametric Techniques

LO1

• Nonparametric tests can be wasteful of data if parametric tests are available for use with the data.

• Nonparametric tests are usually not as widely available and well know as parametric tests.

• For large samples, the calculations for many nonparametric statistics can be tedious.

Disadvantages of Nonparametric Statistics

LO1

Branch of the Tree Diagram Taxonomy Inferential Techniques

LO1

Runs Test

• The one-sample runs test is a nonparametric test of randomness

• The runs test examines the number of runs of each of two possible characteristics that sample items may have

• A run is the order or sequence of observations that have a particular (the same) one of the characteristics. For example, the continuous succession of heads in 15 tosses of a coin.– Example with two runs:

H, H, H, H, H, H, H, H, T, T, T, T, T, T, T– Example with fifteen runs:

H, T, H, T, H, T, H, T, H, T, H, T, H, T, H

LO1

• Sample size: n• Number of sample members possessing the first

characteristic: n1

• Number of sample members possessing the second characteristic: n2

• n = n1 + n2

• If both n1 and n2 are ≤ 20, the small sample runs test is appropriate.

Runs Test: Sample Size Consideration

LO1

• Hypothesize– Step 1: The hypotheses– Ho: The observations in the sample generated randomly

– Ha: The observations in the sample not generated randomly

• TEST– Step 2: let n1 be the number of items with one characteristic and n2

be the number of items in the other.– If the total number of items is less than or equal to 20, small sample

runs test appropriate

• Steps 3 and 4: – Set α and critical regions of test

Setting up the Problem

LO1

– Step 5: Set out the sample data in actual format– Step 6: Tally the number of runs in the sample

• Action– Step 7: Decide whether there is sufficient evidence to accept or reject

the null hypothesis – Step 8: Set out business implications

Setting Out the Problem Continued

LO1

Canadian Tire Store Problemsmall runs test

Step 1: The hypotheses– Ho: The observations in the sample generated randomly

– Ha: The observations in the sample not generated randomly

Step 2: n1= 7, n2 = 8 Step 3: let α=0.05

Step 4: With n1 = 7 and n2 = 8, Table A.11 yields a critical value of 4 and Table A.12 yields a critical value of 13.

* If there are 4 or fewer runs, or 13 or more runs, the decision rule is, reject the null hypothesis.* If the observed runs are between 4 and 13, then the decision rule is, do not reject the null hypothesis.

LO1

Runs Test: Cola Example

LO1

Runs Test: Small Sample Example

Excel cannot analyze data by using the runs test; however, Minitab can. Figure 17.2 is the Minitab output for the cola example runs test. Notice that the output includes the number of runs, 12, and the significance level of the test. For this analysis, diet cola was coded as a 1 and regular cola as a 2. The Minitab runs test is a two-tailed test and the reported significance of the test is equivalent to a p value. Because the significance is 0.9710, the decision is to not reject the null hypothesis.

LO1

• A machine occasionally produces parts that are flawed. • When the machine is working in adjustment, flaws still occur

but seem to happen randomly. A quality control person selects 50 of the parts produced by the machine today and examines them one at a time in the order that they were made. The result is 40 parts with no flaws; and 10 parts with flaws. The following sequence is observed, N= no flaws; F= Flaws:

NNNFNNNNNNNFNNFFNNNNNNFNNN NFNNNNNNFFFFNNNNNNNNNNNN

• The quality controller wishes to determine if the flaws are occurring randomly

Large Sample Machine Problem

LO1

• When samples are large, they start looking like samples that come from normal distributions

• Sampling distribution of R for large samples is approximately normally distributed with a mean and standard deviation of:

• The test statistic is a z statistic computed as:

Points of Interest

LO1

Runs Test: Large Sample Example

LO1

Runs Test: Large Sample Example

LO1

Runs Test: Large Sample Example Minitab Output

LO1

Runs Test: Large Sample Example Minitab Output

LO1

• Mann-Whitney U tests is a nonparametric counterpart of the t test used to compare the means of two independent populations.

• It does not require normally distributed populations

• The assumptions of the model:– The samples are independent.– The level of data is at least ordinal.

Mann-Whitney U Test

LO2

• Let size of sample one be n1

• Let size of sample two be n2

• Small sample case:• If both n1 ≤ 10 and n2 ≤ 10, the small sample procedure is

appropriate.

• Large sample case:• If either n1 or n2 is greater than 10, the large sample procedure is

appropriate.

Mann-Whitney U Test: Sample Size Consideration

LO2

• Arbitrarily designate the two samples as group 1 and group 2. • The data from the two groups are combined into one group,

with each data value retaining a group identifier of its original group.

• The pooled values are then ranked from 1 to n with the smallest number being assigned a rank 1

• Calculate W1 = the sum of the ranks of values from group 1; and W2 = the sum of the ranks of values from group 2.

Calculations of the U-Test

LO2

• Calculating the U statistic for W1 and W2

Mann-Whitney U-Formulas: Small Sample Case

1 11 1 2 1

2 22 1 2 2

'

( 1)

2

( 1)

2

The test statistic is the smallest of these two U values

Bothe u values do not have to be calculated. One can be derived from

the other by the transformation:

n nU n n W

n nU n n W

U

1 2

n n U

LO2

Step 1:

H0: The health service population is identical to the educational service population with respect to employee compensation

Ha: The health service population is not identical to the educational service population with respect to employee compensation

Mann-Whitney U Test: Difference between Health Service Workers and Educational Service Workers

Small Sample Example - Problem 17.1

LO2

Mann-Whitney U Test: Small Sample Example - Demonstration Problem 17.1

Step 2: Because we cannot be certain the populations are normally distributed, we choose a nonparametric alternative to the t test for independent populations: the small-sample Mann-Whitney U test.

Step 3: = .05

Step 4: If the final p-value < .05, reject H0.

Step 5. The sample data are provided.

LO2

Mann-Whitney U Test: Small Sample Example - Demonstration Problem 17.1

LO2

Step 6:• W1 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 =

31• W2 = 5 + 9 + 10 + 11 + 12 + 13 + 14

+ 15 = 89

Because U2 is the smaller value of U, we use Uo = 3 as the test statistic for Table A.13. Because it is the smallest size, let n1 = 7 and n2 = 8.

Mann-Whitney U Test: Small Sample Example - Demonstration Problem 17.1

Step 7: Table A.13 yields a p value of 0.0011. Because this test is two-tailed, we double the p value, producing a final p value of 0.0022. Because the p value is less than α = 0.05, the null hypothesis is rejected. The statistical conclusion is that the populations are not identical.

LO2

Mann-Whitney U Test: Small Sample Example - Demonstration Problem 17.1 – Minitab Output

p value = .0011 from table A.13 and p value = .0046 from minitab.

The difference in p values is due to rounding error in the table.

LO2

For large sample sizes, the value of U is approximately normally distributed. Using an average expected U value for groups of this size and a standard deviation of U’s allows computation of a z score for the U value.

Mann-Whitney U Test: Formulas for Large Sample Case

LO2

Step 1:Ho: The incomes for CBC viewers and non-CBC

viewers are identicalHa: The incomes for CBC viewers and non-CBC

viewers are not identicalStep 2: Use the Mann-Whitney U test for large

samplesSteps 3, 4, and 5:

Incomes of CBC and Non-CBC Viewers

LO2

CBC and Non-CBC Viewers: Calculation of U

LO2

Step 6:

Ranks of Income from Combined Groups of CBC and Non-CBC Viewers

LO2

Step 6:

CBC and Non-CBC Viewers: Conclusion

Step 6:

Step 7:

LO2

Step 8: The fact that CBC viewers have higher average income can affect the type of programming on CBC in terms of both trying to please present viewers and offering programs that might attract viewers of other income levels. In addition, advertising can be sold to appeal to viewers with higher incomes.

• Note that the Mann-Whitney U test cannot be applied to two samples that are related

• Instead, the Wilcoxon Matched-Pairs is applicable to related samples: it is a nonparametric alternative to the t test for related samples

• Applicable to studies in which the data in one group is related to the data in the other group, including before and after studies

• Studies in which measures are taken on the same person or object under different conditions

• Studies of twins or other relatives

Wilcoxon Matched-Pairs Signed Rank Test

LO3

• Differences of the scores of the two matched samples• Differences are ranked, ignoring the sign• Ranks are given the sign of the difference• Positive ranks are summed• Negative ranks are summed• T is the smaller sum of ranks

Wilcoxon Matched-Pairs Signed Rank Test

LO3

• n is the number of matched pairs• If n > 15, T is approximately normally distributed, and a Z test

is used.• If n is small, a special “small sample” procedure is followed:

– The paired data are randomly selected.– The underlying distributions are symmetrical.

• In such a case a critical value against which to compare T is found in Table A.14 of this text.

Wilcoxon Matched-Pairs Signed Rank Test: Sample Size Consideration

LO3

Step 1:

H0: Md = 0

Ha: Md 0

Step 2: n = 6

Step 3: =0.05

Step 4:

If Tobserved 1, reject H0.

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example

LO3

Step 5:

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example

T

Family Pair Toronto Montreal d Rank

1 1,950 1,760 1902 1,840 1,870 -303 2,015 1,810 2054 1,580 1,660 -805 1,790 1,340 4506 1,925 1,765 160

+4-1+5-2+6+3

T = minimum(T+, T-)T+ = 4 + 5 + 6 + 3= 18T- = 1 + 2 = 3T = 3 T = 3 > Tcrit = 1, do not reject H0.

LO3

Step 6:

Step 7:

Wilcoxon Matched-Pairs Signed Rank Test: Small Sample Example

T

Family Pair Toronto Montreal d Rank

1 1,950 1,760 1902 1,840 1,870 -303 2,015 1,810 2054 1,580 1,660 -805 1,790 1,340 4506 1,925 1,765 160

+4-1+5-2+6+3

T = minimum(T+, T-)T+ = 4 + 5 + 6 + 3= 18T- = 1 + 2 = 3T = 3 T = 3 > Tcrit = 1, do not reject H0.

LO3

Step 6:

Step 7:

Wilcoxon Matched-Pairs Signed Rank Test: Minitab Output

STEP 8. Not enough evidence is provided to declare that Toronto and Montreal differ in annual household spending on movie rentals. This information may be useful to movie rental services and stores in the two cities.

p value = 0.142 > α = 0.05, do not reject Ho

LO3

Wilcoxon Matched-Pairs Signed Rank Test: The Large Sample Formulas

LO3

Comparing Airline Cost per Mile of Airfares for 17 Cities in Canada for Both 1979 and 2009

LO3

Airline Cost Comparison: T Calculation

LO3

Airline Cost Comparison: Action

There is no significant difference in the cost of airline tickets between 1979 and 2009.

LO3

• A nonparametric alternative to one-way analysis of variance (ANOVA)

• Like the one-way ANOVA it is used to determine whether c ≥ 3 samples come from the same or different populations

• May be used to analyze ordinal data• It requires and makes no assumption about the

distribution or shape of the population• It assumes that the c groups are independent• It assumes random selection of individual items

Kruskal-Wallis Test

LO4

• The hypotheses tested by the Kruskal-Wallis test follows – Ho: The c populations are identical

– Ha: At least one of the c populations is different

• The process of computing a Kruskal-Wallis K statistic begins with ranking the data in all groups together, as though they were from one group. Beginning with 1 assigned to the smallest value, and so on. Ties are each given the average of the rank of the two values.

• Unlike one-way ANOVA, in which the raw data is analyzed, the Kruskal-Wallis test analyzes the ranks of the data.

The Formulation of the Hypotheses

LO4

Kruskal-Wallis K Statistic

LO4

wherec = number of groups n = total number of items Tj = total of ranks in a group nj = number of items in a group K ≈ χ2, with df = c − 1

Ho: The c populations are identical.

Ha: At least one of the c populations is different.

Number of Office Patients per Physician in Three Organizational Categories

LO4

Patients per Physician Data: Kruskal-Wallis Preliminary Calculations

LO4

Patients per Day Data: Kruskal-Wallis Calculations and Conclusion

LO4

• A nonparametric alternative to the randomized block design• Assumptions– The blocks are independent.– There is no interaction between blocks and treatments.– Observations within each block can be ranked.

• Hypotheses– Ho: The treatment populations are equal

– Ha: At least one treatment population yields larger values than at least one other treatment population

Friedman Test

LO5

Friedman Test

LO5

Ho: The supplier populations are equal

Ha: At least one supplier population yields larger values than at least one other supplier population

Step 1 of Friedman Test: Tensile Strength of Plastic Housings

LO5

Steps 2.3.and 4 of Friedman Test: Tensile Strength of Plastic Housings

LO5

Step 5 of Friedman Test: Tensile Strength of Plastic Housings

LO5

Steps 6 and 7 of Friedman Test: Tensile Strength of Plastic Housings

LO5

• Step 7: Because the observed value of χr2 = 10.68 is greater

than the critical value (7.8147) of chi-square at α = 0.05 , df = 3, the decision is to reject the null hypothesis

• Step 8: The business implication is that statistically, there is a significant difference in the tensile strength of housings made by different suppliers. – Moreover the sample reveals that supplier 3 is producing housings with

a lower tensile strength than those made by other suppliers and that supplier 4 is producing housings with higher tensile strength

– Further study by managers and a quality team may result in attempts to bring supplier 3 up to standard on tensile strength or perhaps cancellation of the contract.

Steps 7 and 8: Action and Business Implications

LO5

Distribution for Tensile Strength Example Figure 17.8

LO5

Friedman Test: Tensile Strength of Plastic Housings – Minitab Output

LO5

• To measure the degree of association of two variables.• When only ordinal-level data or ranked data are available,

Spearman’s rank correlation, rs, can be used to analyze the degree of association of two variables. Charles E. Spearman (1863-1945) developed this correlation coefficient.

Spearman’s Rank Correlation

LO6

Spearman’s Rank Correlation for Heifer and Lamb Prices

LO6

Spearman’s Rank Correlation for Heifer and Lamb Prices

LO6

From the previous slide, the lamb prices are ranked and the heifer prices are ranked. The difference in ranks is computed for each year. The differences are squared and summed, producing Σd2 = 108. The number of pairs, n, is 10. The value of rs = 0.345 indicates that there is a very modest positive correlation between lamb and heifer prices.

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