SW388R7Data Analysis

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Slide 1

Assumption of normality

Assumption of normality

Transformations

Assumption of normality script

Practice problems

SW388R7Data Analysis

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Slide 2

Assumption of Normality

Many of the statistical methods that we will apply require the assumption that a variable or variables are normally distributed.

With multivariate statistics, the assumption is that the combination of variables follows a multivariate normal distribution.

Since there is not a direct test for multivariate normality, we generally test each variable individually and assume that they are multivariate normal if they are individually normal, though this is not necessarily the case.

SW388R7Data Analysis

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Slide 3

Assumption of Normality:Evaluating Normality

There are both graphical and statistical methods for evaluating normality. Graphical methods include the histogram

and normality plot. Statistical methods include diagnostic

hypothesis tests for normality, and a rule of thumb that says a variable is reasonably close to normal if its skewness and kurtosis have values between –1.0 and +1.0.

None of the methods is absolutely definitive. We will use the criteria that the skewness

and kurtosis of the distribution both fall between -1.0 and +1.0.

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Slide 4

Assumption of Normality:Histograms and Normality Plots

RS OCCUPATIONAL PRESTIGE SCORE (1980)

85.0

80.0

75.0

70.0

65.0

60.0

55.0

50.0

45.0

40.0

35.0

30.0

25.0

20.0

15.0

Histogram

Fre

quency

50

40

30

20

10

0

Std. Dev = 13.94

Mean = 44.2

N = 255.00

TIME SPENT USING E-MAIL

40.035.030.025.020.015.010.05.00.0

Histogram

Fre

quen

cy

100

80

60

40

20

0

Std. Dev = 6.14

Mean = 3.6

N = 119.00

Normal Q-Q Plot of TIME SPENT USING E-MAIL

Observed Value

50403020100-10

Expecte

d N

orm

al

3

2

1

0

-1

-2

Normal Q-Q Plot of RS OCCUPATIONAL PRESTIGE SCORE (1980)

Observed Value

100806040200

Exp

ecte

d N

orm

al

3

2

1

0

-1

-2

-3

On the left side of the slide is the histogram and normality plot for a occupational prestige that could reasonably be characterized as normal. Time using email, on the right, is not normally distributed.

SW388R7Data Analysis

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Slide 5

Assumption of Normality:Hypothesis test of normality

Tests of Normality

.121 255 .000 .964 255 .000RS OCCUPATIONALPRESTIGE SCORE (1980)

Statistic df Sig. Statistic df Sig.

Kolmogorov-Smirnova

Shapiro-Wilk

Lilliefors Significance Correctiona.

Tests of Normality

.296 119 .000 .601 119 .000TIME SPENTUSING E-MAIL

Statistic df Sig. Statistic df Sig.

Kolmogorov-Smirnova

Shapiro-Wilk

Lilliefors Significance Correctiona.

The hypothesis test for normality tests the null hypothesis that the variable is normal, i.e. the actual distribution of the variable fits the pattern we would expect if it is normal. If we fail to reject the null hypothesis, we conclude that the distribution is normal.

The distribution for both of the variable depicted on the previous slide are associated with low significance values that lead to rejecting the null hypothesis and concluding that neither occupational prestige nor time using email is normally distributed.

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Slide 6

Assumption of Normality:Skewness, kurtosis, and normality

Using the rule of thumb that a rule of thumb that says a variable is reasonably close to normal if its skewness and kurtosis have values between –1.0 and +1.0, we would decide that occupational prestige is normally distributed and time using email is not.

We will use this rule of thumb for normality in our strategy for solving problems.

SW388R7Data Analysis

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Slide 7

Transformations

When a variable is not normally distributed, we can create a transformed variable and test it for normality. If the transformed variable is normally distributed, we can substitute it in our analysis.

Three common transformations are: the logarithmic transformation, the square root transformation, and the inverse transformation.

All of these change the measuring scale on the horizontal axis of a histogram to produce a transformed variable that is mathematically equivalent to the original variable.

SW388R7Data Analysis

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Slide 8

When transformations do not work

When none of the transformations induces normality in a variable, including that variable in the analysis will reduce our effectiveness at identifying statistical relationships, i.e. we lose power to detect relationship and estimated values of the dependent variable based on our analysis may be biased or systematically incorrect.

We do have the option of changing the way the information in the variable is represented, e.g. substitute several dichotomous variables for a single metric variable.

SW388R7Data Analysis

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Slide 9

Problem 1

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Slide 10

Computing “Explore” descriptive statistics

To compute the statistics needed for evaluating the normality of a variable, select the Explore… command from the Descriptive Statistics menu.

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Slide 11

Adding the variable to be evaluated

First, click on the variable to be included in the analysis to highlight it.

Second, click on right arrow button to move the highlighted variable to the Dependent List.

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Slide 12

Selecting statistics to be computed

To select the statistics for the output, click on the Statistics… command button.

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Slide 13

Including descriptive statistics

First, click on the Descriptives checkbox to select it. Clear the other checkboxes.

Second, click on the Continue button to complete the request for statistics.

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Slide 14

Selecting charts for the output

To select the diagnostic charts for the output, click on the Plots… command button.

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Slide 15

Including diagnostic plots and statistics

First, click on the None option button on the Boxplots panel since boxplots are not as helpful as other charts in assessing normality.

Second, click on the Normality plots with tests checkbox to include normality plots and the hypothesis tests for normality.

Third, click on the Histogram checkbox to include a histogram in the output. You may want to examine the stem-and-leaf plot as well, though I find it less useful.

Finally, click on the Continue button to complete the request.

SW388R7Data Analysis

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Slide 16

Completing the specifications for the analysis

Click on the OK button to complete the specifications for the analysis and request SPSS to produce the output.

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Slide 17

The histogram

An initial impression of the normality of the distribution can be gained by examining the histogram.

In this example, the histogram shows a substantial violation of normality caused by a extremely large value in the distribution.

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Slide 18

The normality plot

The problem with the normality of this variable’s distribution is reinforced by the normality plot.

If the variable were normally distributed, the red dots would fit the green line very closely. In this case, the red points in the upper right of the chart indicate the severe skewing caused by the extremely large data values.

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Slide 19

Tests of Normality

.246 93 .000 .606 93 .000TOTAL TIME SPENTON THE INTERNET

Statistic df Sig. Statistic df Sig.

Kolmogorov-Smirnova

Shapiro-Wilk

Lilliefors Significance Correctiona.

The test of normality

If problem 1 had asked about the results of the test of normality, we would look at that output for the evaluation. Since the sample size is larger than 50, we use the Kolmogorov-Smirnov test. If the sample size were 50 or less, we would use the Shapiro-Wilk statistic instead.

The null hypothesis for the test of normality states that the actual distribution of the variable is equal to the expected distribution, i.e., the variable is normally distributed. Since the probability associated with the test of normality is < 0.001 is less than or equal to the level of significance (0.01), we reject the null hypothesis and conclude that total hours spent on the Internet is not normally distributed. (Note: we report the probability as <0.001 instead of .000 to be clear that the probability is not really zero.)

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Slide 20

Descriptives

10.731 1.5918

7.570

13.893

8.295

5.500

235.655

15.3511

.2

102.0

101.8

10.200

3.532 .250

15.614 .495

Mean

Lower Bound

Upper Bound

95% ConfidenceInterval for Mean

5% Trimmed Mean

Median

Variance

Std. Deviation

Minimum

Maximum

Range

Interquartile Range

Skewness

Kurtosis

TOTAL TIME SPENTON THE INTERNET

Statistic Std. Error

Table of descriptive statistics

To answer problem 1, we look at the values for skewness and kurtosis in the Descriptives table.

The skewness and kurtosis for the variable both exceed the rule of thumb criteria of 1.0. The variable is not normally distributed.

SW388R7Data Analysis

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Slide 21

Problem 1 - Answer

SW388R7Data Analysis

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Slide 22

The assumption of normality script

First, move the variables to the list boxes based on the role that the variable plays in the analysis and its level of measurement.

Third, mark the checkboxes for the transformations that we want to test in evaluating the assumption.

Second, click on the Assumption of Normality option button to request that SPSS produce the output needed to evaluate the assumption of normality.

Fourth, click on the OK button to produce the output.

SW388R7Data Analysis

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Slide 23

Script output for testing normality

The script produces the same output that we computed manually, in this example, the skewness and kurtosis.

However, what makes the script more useful is that it will compute transformations and test for their conformity to the assumption.

SW388R7Data Analysis

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Slide 24

Descriptive statistics for transformed variables

While this problem does not ask about transformations, the next one does, and we will use this output to answer the question.

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Slide 25

Problem 2

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Slide 26

Descriptive statistics for transformed variables

The skewness and kurtosis for each of the transformed variables is shown in the table. The logarithmic transformation produced a variable that satisfies the assumption of normality; the others do not.

SW388R7Data Analysis

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Slide 27

Problem 2 - Answer

SW388R7Data Analysis

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Slide 28

Other problems on assumption of normality

A problem may ask about the assumption of normality for a nominal level variable. The answer will be “An inappropriate application of a statistic” since there is no expectation that a nominal variable be normal.

A problem may ask about the assumption of normality for an ordinal level variable. If the variable or transformed variable is normal, the correct answer to the question is “True with caution” since we may be required to defend treating an ordinal variable as metric.

Questions will specify a level of significance to use and the statistical evidence upon which you should base your answer.

SW388R7Data Analysis

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Slide 29

Steps in answering questions about the assumption of normality – question 1

Question: is the variable normally distributed?

Does the statistical evidence support the normality assumption?

Yes

No

Incorrect application of a statistic

Yes

NoIs the variable to be evaluated metric?

False

Is the variable evaluated ordinal level?

Yes

TrueNo

True with caution

SW388R7Data Analysis

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Slide 30

Steps in answering questions about the assumption of normality – question 2

Statistical evidence supports normality?

Yes

No

Incorrect application of a statistic

Yes

NoIs the variable to be evaluated metric?

Statistical evidence for transformation supports normality?

Is the variable ordinal level?

Yes

False (transformation is not normal)

True (transformation

is normal)

True with caution(transformation

is normal

Question: is the variable NOT normally distributed, but a transformed variable is normally distributed?

Yes

False(distribution is normal) No

No