C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample

C82MCP Diploma Statistics

School of PsychologyUniversity of Nottingham

1

Overview of Lecture

• Parametric vs Non-Parametric Statistical Tests.• Single Sample Chi-Square• Multi-Sample Chi-Square• Analysing Chi-Square Residuals



2

Parametric Vs Non-Parametric Statistical Tests.

• Many statistical tests make assumptions about the population from which the scores are taken.

• The most common assumption is that the data is normally distributed.

• Some statistical tests don't make assumptions about the population from which the scores are taken.



3

Parametric Tests

• Parametric tests test hypotheses about specific parameters such as the mean or the variance.

• They make the assumption that these parameters are central to our research hypotheses.



4

Parametric Test Assumptions.

• Parametric tests usually (thought not always) make the following assumptions:• The scores must be independent. In other words the

selection of any particular score must not bias the chance of any other case for inclusion.

• The observations must be drawn from normally distributed populations.

• The populations (if comparing two or more groups) must have the same variance.

• The variables must have been measured in at least an interval scale so that is is possible to interpret the results.



5

Non-Parametric Tests

• Non-parametric tests on the other hand are based on a statistical model that has only very few assumptions.

• None of these assumptions include making assumptions about the form of the population distribution from which the sample was taken.

• Whenever we look at categorical or ordinal data we usually use non-parametric tests.

• Furthermore, if we can show that the data is not normally distributed we should also use non-parametric tests (but there are exceptions).



6

Nominal/Categorical Scale Data

• Numbers are used to divide different behaviours into different classes without implying that the different classes are numerically related to each other.

• Whenever we look at nominal or categorical data we usually use non-parametric tests

• These non-parametric tests focus on the frequencies or counts of membership of categories or nominal groups



7

Single Sample Chi-Square Statistic - Rationale

• When the Null Hypothesis is true• The observed differences in frequencies will be due to

chance• When the Null Hypothesis is false

• The differences in frequencies will reflect actual differences in the population



8

Single Sample Chi-Square Statistic - Method

• Arrange the data in a table• Each category has a separate entry• The number of members of each category are counted

• Calculate the frequencies expected by chance• Find the difference between the observed & expected

frequencies



9

Single Sample Chi-Square Statistic - Method

Research

Academic

Clinical

Occupational

Educational

Total

Job Observed Frequency

Expected Frequency

Observed - Expected

10

5

30

15

10

70

14

14

14

14

14

-4

-9

16

1

-4

ExpectedTotal Number of CasesNumber of Categories



10

Single Sample Chi-Square Statistic - Formula

• The test statistic, , is calculated by:

• Where is the observed frequency is the expected frequency

2 ( fo fe)

fe

2

fo

fe

2



11

Research

Academic

Clinical

Occupational

Educational

Total

Job Observed Frequency

Expected Frequency

Observed - Expected

10

5

30

15

10

70

14

14

14

14

14

70

-4

-9

16

1

-4

2( 4)214

( 9)2

14 (16)2

14(1)2

14 ( 4)2

14

Expected Frequencies

Observed-Expected Frequencies



12

Single Sample Chi-Square Statistic - Significance• In order to test the null hypothesis that the distribution of

frequencies is equal (i.e. occurred by chance) we look up a critical value of chi-square in tables

• To do this we need to know the degrees of freedom associated with the chi-square• degrees of freedom = number of categories-1

• We reject the null hypothesis when

• For this data we can reject the null hypothesis

observed2 critical

2



13

Single Sample Chi-Square Statistic - Interpretation• Rejecting the null hypothesis

• This means that the frequencies associated with each of the categories did not represent only chance fluctuations in the data

• Failing to reject the null hypothesis• This means that the differences in the frequencies

associated with each of the categories was due to chance fluctuations in the data



14

Multi-Sample Chi-Square Statistic - Rationale

• Used when we look at the relationship between two independent variables and their effects on frequencies

• Under the null hypothesis• The differences in the observed frequencies are due to

chance• When the null hypothesis is false

• The difference in the observed frequencies are due to the effects of the two variables



15

Multi-Sample Chi-Square Statistic - Method

• Arrange the data in a table• Each category has a separate entry• The number of members of each category are counted

• Calculate the frequencies expected by chance• Find the difference between the observed & expected

frequencies



16

Multi-Sample Chi-Square Statistic - Method

Research

Academic

Clinical

Occupational

Educational

Total

Job Females Expected Frequency

Observed - Expected

Males Expected Frequency

Observed - Expected

Total

10

5

30

15

10

70

15

7.5

20

17.5

10

-5

-2.5

10

-2.5

0

20

10

10

20

10

70

15

7.5

20

17.5

10

5

2.5

-10

2.5

0

30

15

40

35

20

140

ExpectedRow Total x Column TotalGrand Total



17

Multi-Sample Chi-Square Statistic - Formula

• The test statistic, , is calculated by:


2

2 ( fo fe)

fe

2

fo

fe



18

Research

Academic

Clinical

Occupational

Educational

Total

Job Females Expected Frequency

Observed - Expected

Males Expected Frequency

Observed - Expected

Total

10

5

30

15

10

70

15

7.5

20

17.5

10

-5

-2.5

10

-2.5

0

20

10

10

20

10

70

15

7.5

20

17.5

10

5

2.5

-10

2.5

0

30

15

40

35

20

140

2( 5)215

( 2.5)2

7.5 (10)2

20......( 10)2

20 (2.5)2

17.5 (0)2

10

Observed-Expected Frequencies

Expected Frequencies



19

Multi-Sample Chi-Square Statistic - Significance

• In order to test the null hypothesis that the distribution of frequencies is equal (i.e. occurred by chance) we look up a critical value of chi-square in tables

• To do this we need to know the degrees of freedom associated with the chi-square• degrees of freedom = (rows-1)(columns-1)

• We reject the null hypothesis when

• For this data we can reject the null hypothesis

observed2 critical

2



20

Multi-Sample Chi-Square Statistic - Interpretation• Rejecting the null hypothesis

• This means that the frequencies associated with cell in the design did not represent only chance fluctuations in the data.

• Failing to reject the null hypothesis• This means that the differences in the frequencies

associated with each cell in the design was due to chance fluctuations in the data



21

Chi-Square Statistic - Analysing Residuals

• Since the Chi-Square Statistic is calculated using all the information from the experiment:• it tell us that at least one of the cell frequencies is

different from chance• it cannot tell which cell frequency is different from chance

• To find out which cells differ from what we would expect by chance we analyse the residuals• residuals - what is left over after we have removed the

effect of chance



22

Analysing Residuals - Formula

• A residual is calculated by:


Residual( fo fe)

fe

fo

fe



23

Analysing Residuals - Interpretation

• When a residual |±1.96|• There is a significance difference between the observed

and expected frequencies



24

Pearson's Chi-Square - Assumptions

• The categories must be mutually exclusive. In other words no single subject can contribute a score to more than one category.

• The observations must be independent. A particular score cannot influence any other score.

• Both the observed and the expected frequencies must be greater than or equal to 5.

Documents

C82MCP Diploma Statistics School of Psychology University of Nottingham 1 Overview of Lecture Parametric vs Non-Parametric Statistical Tests. Single Sample