Analisis Non-Parametrik

Antonius NW PratamaMK Metodologi Penelitian

Bagian Farmasi Klinik dan KomunitasFakultas Farmasi Universitas Jember

Reference

• Bolton, S., Bon, C., 2004, Pharmaceutical Statistics Practical and Clinical Applications, 4th ed, New York: Marcel Dekker, Inc.

Outline

• The probability distribution– Binomial and normal distribution

• Normality assessment• Non-parametric tests

Probability Distribution

• “A probability distribution is a mathematical representation (function) of the probabilities associated with the values of a random variable.”– For discrete, discontinuous, binary data: the

binomial probability distribution– For continuous data: the normal distribution

The Binomial Distribution

The Binomial Distribution

• Two parameters: – the probability of one or the other outcome, – the number of trials or observations, N

• The probability of the outcome of a binomial experiment consisting of N trials:

(p+q)N

» p: the probability of success» q: the probability of failure

The Binomial Distribution

• Three coin tosses possible results: N+1=4– Three heads– Two heads and one tail– Two tails and one head– Three tails

The Binomial Distribution

The Binomial Distribution

The Binomial Distribution

• Three patients receiving antibiotics possible results: N+1=41. Three cures2. Two cures and one

failure3. Two failures and one

cure4. Three failures

The Binomial Distribution

• Four patients receiving antibiotics, p=0.75, q=.25

The Binomial Distribution

The Binomial Distribution

The Normal Distribution

• A continuous probability distribution and consists of an infinite number of values.

• Scientific data from pharmaceutical experiments cannot possibly fit this definition.

• Nevertheless, if real data conform reasonably well with the theoretical definition of the normal curve, adequate approximations, if not very accurate estimates of probability, can be computed based on normal curve theory.

The Normal Distribution

The Normal Distribution

The Normal Distribution

The Normal Distribution

The Normal Distributionstandard normal curve, standard score of Z

Z: the number of SDs from the mean

The Normal Distributionstandard normal curve, standard score of Z

Transformation of a value to a standard score(standardizing):

The Normal Distributionstandard normal curve, standard score of Z

The Normal Distributionstandard normal curve, standard score of Z

Other Common Probability Distributions

• The Poisson distribution• The Student’s t distribution• The chi-square (X2) distribution• The F distribution

Assessing normality

• 2 ways: with statistical tests or without• Without statistical test:– In a statistical package, such as SPSS, look at:

• Histogram symmetric: good• Box plot symmetric: good, outliers?• Normal Q-Q plot straight line: good• Skewness within -1 to 1, close to zero: good • Kurtosis within -1 to 1, close to zero: good (in STATA:

normal value is 3)

• Good results: normally distributed data parametric tests. Give examples!

Data not normally distributed?

• Do transformation to achieve Normality!– natural logarithm i.e. ln(X )– square root i.e. √X– reciprocal i.e. 1/X– square i.e. X 2

• Transforming the data is merely a change of units, like converting pounds to kilograms

• Assess its normality again! Bad results? Use non-parametric tests

Non-Parametric Tests

• Distribution-free statistics• Remember data types: nominal, ordinal,

interval (no true zero), ratio• Most efficient for nominal or ordinal (ranked)

data • Appropriate for continuous data not normally

distributed

Non-Parametric Tests

• However, non-parametric tests can be applied to most of the data that we usually encounter, including that from continuous data distributions.

• Hence, data that are normally distributed may also be analyzed using these methods.– Disadvantage: less sensitive

Non-Parametric Tests

Non-Parametric Tests

Non-Parametric Tests

• Basically, converting data into ranks• Sign test, the paired t-test alternative• Mann-Whitney test, alternative to the two sample t-

test• Wilcoxon paired samples test, a substitute for the

paired t-test• Kruskal-Wallis test, alternative to the one-way ANOVA• Spearman’s rho rank-sum correlation, a substitute for

Pearson’s coefficient correlation

Sign Test

• Probably the simplest• The sign test is a test of the equality of the

medians of two comparative groups.• Paired data• The pairs are matched, and differences of the

measurements for each pair tabulated.• Ties (zero difference) are not counted.

Sign Test

Wilcoxon Signed Rank Test

• More powerful than sign test as it takes the magnitude of the differences into consideration

• Ties, again, are not counted.

Wilcoxon Signed Rank Test

Wilcoxon Signed Rank Test

Wilcoxon rank sum test (Mann-Whitney U-test)

• Alternative to the two independent sample t-test

• What is the meaning of “independent” here?

Contingency Tables

• Chi-square tests for contingency tables (e.g., 2x2 tables) are often categorized as non-parametric tests.

• nominal or categorical data which cannot be analyzed using the ranking techniques discussed above.

• These data cannot be ordered (the data are not ordinal or on an interval/ratio scale)

• Fisher’s exact test

Contingency Tables

Contingency Tables

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