Non-Parametric Tests

Question 1

Problems with parametric tests

Answer 1

Strong assumptions e.g. Normality e.g. N large enough to invoke CLT.

Question 2

What are non-parametric tests?

Answer 2

Tests valid over a wide range of distributions and can be carried out making far fewer assumptions about the random variable.

Question 3

What is the most simple non parametric test called?

Answer 3

Wilcoxon Sign test

Question 4

What type of data does the sign test analyse?

Answer 4

Matched pairs

Question 5

Briefly describe the process of setting up sign test

Answer 5

Assign + if 1st value > 2nd
- if 1st value < 2nd
Construct a Bernoulli trial for each individual
Under H0, p=0.5. Repeated Bernoulli = binomial
W ~ B(n, 0.5). P (W is what we observe)

Question 6

For sign test how do we calculate the p value if the test is two sided?

Answer 6

We work out probability using binomial e.g. 9C0 (0.5)^9 + 9C1 (0.5)^9 = 0.02
P value for 2 sided test = 2 x 0.02 = 0.04
0.04 < 0.05 (alpha) therefore we reject H0

Question 7

For a sign test if n>25, how do we work out the probability?

Answer 7

Invoke CLT so W bar is approx normal with M=0.5 and sigma^2 = 0.5^2 / n.

Z = (W/n - 0.5) / (sqrt 0.5^2 / n)

Question 8

Problem with sign test

Answer 8

Ignores magnitude - treats a large negative difference the same as a small negative difference. Collapse everything to 0 or 1 = lots of information thrown away = low powered test when n is small. More likely to make a type 2 error of accepting H0 when it’s false. So we often find an insignificant test statistic.

Question 9

What do we do with zero differences in the sign test?

Answer 9

We discard them and then reduced n by 1.

Question 10

The sign test & sign rank test are only applicable for…

Answer 10

Matched pairs

Question 11

How does the sign rank test differ from sign test?

Answer 11

It accounts for magnitude of the difference as well as sign

Question 12

Describe sign rank test

Answer 12

Rank absolute differences in ascending order of magnitude
If two values have the same magnitude, assign the average rank
Sum up R+ and R- separately

Question 13

What is our test statistic for the sign rank test?

Answer 13

T = MIN {R+, R-}

Question 14

Under H0 for the sign rank test, what is E(T) and V(T)

Answer 14

E(T) = n(n+1)/4
V(T) = n(n+1)(2n+1)/24

Question 15

What n<25, how do we work out our CVs for the sign rank test?

Answer 15

Use the tables given in the formula sheet. Correct value of alpha dependent on 1/2/ sided test.

Question 16

When do we reject H0 for the sign rank test? Why?

Answer 16

If our test statistic < CV

As we are minimising

Question 17

If n>25, what do we do for the sign rank test?

Answer 17

Invoke CLT = approx normality

Our test statistic is given by [T - n(n+1)]/4 / sqrt [n(n+1)(2n+1)/24]

Question 18

Limitation of sign rank test

Answer 18

Ignores spread of data - if highest absolute difference is 2, given rank n. If highest is 100, still given rank n. This may compress or stretch data. Less powerful than a parametric test, but more than just the sign test.

Question 19

When n>25 for sign rank test, when do we reject?

Answer 19

Reject if p value is less than the significance level (same as usual hypothesis testing)

Question 20

When is the Mann-Whitney test applicable?

Answer 20

We can use it even if we don’t have matched pairs. Use for independent random samples for difference in means.

Question 21

How do we rank equal magnitudes in the sign rank test?

Answer 21

Average rank e.g. If two numbers are to be ranked 4&5, give the, both rank 4+5/2 = 4.5

Question 22

Describe the Mann Whitney test

Answer 22

Rank all observations n1 + n2 but preserve the colour
Equal values given an average rank
Sum of R1
Work out U(see formula sheet), E(U) & V(U) and then test statistic
If n>25, approx normal.

Question 23

When do we reject for Mann Whitney test?

Answer 23

If n>25, approx normal = Z test
Double p value
Reject if p value is less than the significant level

Question 24

When do we use goodness of fit test?

Answer 24

Where we have discrete outcomes into k categories (can also use for continuous data but need to put into discrete categories first)

Question 25

Describe goodness of fit test

Answer 25

Calculate Ei = npi for each category K
See formula sheet to calculate test statistic using simpler version
Follows a chi squared distribution.
Reject if test stat > CV given by chi squared

Question 26

When do we reject H0 for goodness of fit test?

Answer 26

If the test statistic is greater than the CV given by chi squared distribution

Question 27

What distribution do we get the CV from for a goodness of fit test?

Answer 27

Chi squared

Question 28

DOF for CV for goodness of fit test =

Answer 28

DOF = K - 1 where K is the number of different categories

Question 29

What is a condition for the goodness of fit test to be appropriate? How can we solve it?

Answer 29

Ei should not be <5 for any category; if it is aggregate two categories.

Question 30

What is H0 usually for goodness of fit test ?

Answer 30

H0 = all outcomes equally likely
So Pi = 1/k
Ei = n/k for each category

Question 31

What data are contingency tables used for?

Answer 31

Where we have a two way table with K categories in A and H in B, so we have KH cross classifications.

Question 32

Why don’t we use hypothesis testing or ANOVA instead of contingency tables?

Answer 32

Hypothesis tests limited to two groups

ANOVA allows >2 groups but requires assumption of normality.

Question 33

What are contingency tables another form of?

Answer 33

Goodness of fit test but with a two way table rather than one

Question 34

What is H0 for contingency table?

Answer 34

H0 = variables are not related 
H1 = variables are related

Question 35

How do we work out the expected values for contingency tables?

Answer 35

Eij = n pij

Since under H0 variables are independent, pij = p(i) x p(j)

Question 36

What distribution do we get CVs from for contingency tables?

Answer 36

Chi squared

Question 37

How do we work out degrees of freedom for contingency tables?

Answer 37

DOF = (r - 1)(c - 1)

r=Number of rows
c=number of columns

Question 38

When do we reject H0 for contingency tables?

Answer 38

If test statistic is greater than CV given by chi squared distribution