Four levels of measurement

Nominal

Ordinal

Interval

Ratio

Nominal: the lowest level

Ordinal

Interval

Ratio: the highest level

Classic Parametric Setting

For the classic ANOVA, we assume we are sampling from normally distributed populations with identical variances. From here we test the hypothesis:

H(0):μ1= μ2 = μ3 (All distributions are identical) or μ1- μ overall = 0, μ2 - μ overall = 0, μ3 - μ overall =0

Nonparametric Setting

H(0): F0 = F2 = F4 = F6 = F8 (Identical Distributions) orF0 - H = 0, F2 - H = 0, F4 - H = 0, F6 - H = 0, F8 - H = 0where H is the overall average distribution.

Settings for Nonparametric Tests

• When data are very non-normal (e.g. very skewed, or presence of outliers)

• If you suspect the variable is not normally distributed in the population

• When you have ordinal data and are interested in comparing distributions and their relationships.

• If other assumptions of parametric tests are violated (e.g. homogeneity of variance, ordinal

• data)

Nonparametric Methods

• Can provide invariant results when strictly monotone transformations of the data are used.

Gamma GT StudyNote Non Constant Variance, Outliers, and Heavily Skewed Distributions

Gamma GT MeansOriginal data means, means of ln(x), and means of ln(ln(x)). Note reversal of treatment means. Nonparametric gives same result for all 3 ‘versions’ of the data

Nonparametric Methods

• Absence of variability in treatment groups is admitted

Panic 2 Study. Note skewed distributions, outliers, andzero variance at Week 10.

Nonparametric Methods

• Can provide approximations for small sample sizes

Nonparametric Methods

• Can be performed using SAS macros

• “Proc Mixed”

Disadvantages - Advantages

• Lesser power relative to corresponding parametric test (when assumptions of the parametric test are met)

• When assumptions of parametric tests are not met nonparametric methods can be more powerful

Definition of Probability ‘Distributions’

Probability Distribution Function (f): A function that assigns a probability to each possible outcome.

P(X = Some particular set of values of x) for all possible x

Cumulative Probability Distribution Function (F): A function that assigns a probability to a value less than or equal to each possible outcome.

P(X some particular value of x) for all possible x

Empirical Distribution Function F

If we have a sample from a ‘Parent Distribution’ F, our estimate of F from the sample is the Empirical Distribution Function, denoted F hat.

iF̂

Panic1 Study Empirical Normalized Dist. Functions

Relative Effects P(i)The empirical distribution functions F(i), do provide information to detect differences among the different group distributions.

But there exists a summary measure that describes thelikelihood that values from one distribution tend to be greater or lesser than the overall mean distribution H. This measure is called:

the ‘Relative Effect’ for group i.

Interpretation of Relative Effects p(i)

The relative effect p(i) quantifies the tendency of the marginal distribution F(i) with respect to the overall average distribution H.

If F(i) tends to lie in the region to the left of H then p(i) < 0.5.

If F(i) tends to lie in the region to the right of H then p(i) > 0.5.

Calculation of Estimated Relative Effects: A function of the ranks

The Relationship of the Distribution function F(i), p(i),

and μ (i)

a

aP

aF

H

and

pppH

FFFH

...:

...:

...:

210

210

210

Covariance Matrix Notes:

1. A covariance matrix must have an inverse in orderto calculate many multivariate stats (Wald, Hotelling).

2. A covariance matrix of a set of variables in which onevariable has ZERO variance will not have an inverse.

3. Both of the above types of covariance matrices have a TRACE, the sum of the diagonal elements.

Row t0 t2 t4 t6 t8

1 8 6 5 5 4 2 8 6 5 4 2 3 6 5 5 4 2 4 6 6 6 5 5 5 7 6 6 6 6 6 8 7 3 2 2 7 7 6 7 3 3 8 6 4 5 3 3 9 5 4 3 3 2 10 8 6 5 5 4 11 7 6 5 4 2 12 6 5 5 4 2 13 6 6 6 5 5 14 8 6 6 6 6 15 8 7 4 2 2 16 7 6 7 3 3

Row t0 t2 t4 t6 t8

1 8 6 5 5 4 2 8 6 5 4 2 3 6 5 5 4 2 4 6 6 6 5 5 5 7 6 6 6 6 6 8 7 3 2 2 7 7 6 7 3 3 8 6 4 5 3 3 9 5 4 3 3 2 10 8 6 5 5 4 11 7 6 5 4 2 12 6 5 5 4 2 13 6 6 6 5 5 14 8 6 6 6 6 15 8 7 4 2 2 16 7 6 7 3 3

Panic1 Study Empirical Normalized Dist. Functions

Relative Effects P(i)The empirical distribution functions F(i), do provide information to detect differences among the different group distributions.

But there exists a summary measure that describes thelikelihood that values from one distribution tend to be greater or lesser than the overall mean distribution H. This measure is called:

the ‘Relative Effect’ for group i.

Interpretation of Relative Effects p(i)

The relative effect p(i) quantifies the tendency of the marginal distribution F(i) with respect to the overall average distribution H.

If F(i) tends to lie in the region to the left of H then p(i) < 0.5.

If F(i) tends to lie in the region to the right of H then p(i) > 0.5.

Calculation of Estimated Relative Effects: A function of the ranks

The Relationship of the Distribution function F(i), p(i),

and μ (i)

a

aP

aF

H

and

pppH

FFFH

...:

...:

...:

210

210

210

Panic1 Empirical Normalized Distributions F(i)

Panic1 Estimated Relative Effects for P(i) and 95% CI’s

data panic1;input patient t1-t5;array t{5} t1-t5;do i=1 to 5; week=(i-1)*2; score=t{i};output; end;drop i t1-t5;datalines;1 8 6 5 5 4 2 8 6 5 4 2 3 6 5 5 4 2 4 6 6 6 5 5 5 7 6 6 6 6 6 8 7 3 2 2 7 7 6 7 3 3 8 6 4 5 3 3 9 5 4 3 3 2 10 8 6 5 5 4 11 7 6 5 4 2 12 6 5 5 4 2 13 6 6 6 5 5 14 8 6 6 6 6 15 8 7 4 2 2 16 7 6 7 3 3 ;run;

Obs patient week score  1 1 0 8 2 1 2 6 3 1 4 5 4 1 6 5 5 1 8 4 6 2 0 8 7 2 2 6 8 2 4 5 9 2 6 4 10 2 8 2 . . . . . . . . . . . . 71 15 0 8 72 15 2 7 73 15 4 4 74 15 6 2 75 15 8 2 76 16 0 7 77 16 2 6 78 16 4 7 79 16 6 3 80 16 8 3

%include 'E:\nonparametric\SAS Macros\ld_f1.sas';

%ld_f1(data=panic1, var=score, time=week, subject=patient, var_pt=_no_, time_pt=_no_); run;

%include 'E:\nonparametric\SAS Macros\ld_ci.sas';

%ld_ci(data=panic1, var=score, time=week, subject=patient, alpha=0.05); run;

LD_1F --- subjects x T

T: fixed, subjects: random

SAS-data-filename: panic1

response-variable: score

Class Level Information

CLASS LEVELS

T WEEK 5

Total number of observations 80

Total number of subjects 16

Number of missing values 0

RTE = relative treatment Effects

Nobs = Number of observations (do not count

the repeated measurements within the cells)

Source Time Rank means Nobs RTE

week 0 66.094 16 .81992

week 2 50.500 16 .62500

week 4 41.281 16 .50977

week 6 25.625 16 .31406

week 8 19.000 16 .23125

The SAS System 10

17:29 Friday, October 3, 2003

----------------------------------------------------------

Statistics df1 df2 p-value

Wald Chi^2(df1) 126.69 4.0000 . .00000

Hotelling F(df1,df2) 25.339 4.0000 12.000 .00001

Anova Chi^2(df1)/df1 36.937 2.2341 . .00000

The SAS System 11

17:29 Friday, October 3, 2003

LD_CI

Bias-Estimation and Confidence-Intervals for Relative Effects

SAS-Data-Filename: panic1

Response-Variable: score Observations: 80

Group-Variable: _none_ Groups: 1

Time-Variable: week Timepoints: 5

Subject-Variable patient Subjects: 16

Relative Effects, Biases, Variances and Confidence-Limits (alpha=0.05)

Time RE Bias Variance lower upper

0 .81992 .00284 .00768 .76566 .85379

2 .62500 -.0008 .01571 .56079 .68278

4 .50977 .00310 .02512 .43271 .58610

6 .31406 -.0018 .00800 .27316 .36061

8 .23125 -.0033 .01597 .18033 .30524