Wednesday, May 28, 2014

Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication

This is one of the following seven articles on Two-Factor ANOVA With Replication in Excel

Two-Factor ANOVA With Replication in 5 Steps in Excel 2010 and Excel 2013

Variance Tests: Levene’s and Brown-Forsythe For 2-Factor ANOVA in Excel 2010 and Excel 2013

Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication Effect Size in Excel 2010 and Excel 2013

Excel Post Hoc Tukey’s HSD Test For 2-Factor ANOVA With Replication

2-Factor ANOVA With Replication – Test Power With G-Power Utility

Scheirer-Ray-Hare Test Alternative For 2-Factor ANOVA With Replication

Shapiro-Wilk Normality Test in Excel For 2-Factor ANOVA With Replication

There are a number of normality test that can be performed on each group’s data. The normality test that is preferred because it is considered to be more powerful (accurate) than the others, particularly with smaller sample sizes is the Shapiro-Wilk test.

The Shapiro-Wilk Test is a hypothesis test that is widely used to determine whether a data sample is normally distributed. A test statistic W is calculated. If this test statistic is less than a critical value of W for a given level of significance (alpha) and sample size, the Null Hypothesis which states that the sample is normally distributed is rejected.

The Shapiro-Wilk Test is a robust normality test and is widely-used because of its slightly superior performance against other normality tests, especially with small sample sizes. Superior performance means that it correctly rejects the Null Hypothesis that the data are not normally distributed a slightly higher percentage of times than most other normality tests, particularly at small sample sizes.

The Shapiro-Wilk normality test is generally regarded as being slightly more powerful than the Anderson-Darling normality test, which in turn is regarded as being slightly more powerful than the Kolmogorov-Smirnov normality test.

Here is a summary of the results of the Shapiro-Wilk normality test performed on the sample groups that constitute each of the levels of each of the two factors.

The Shapiro-Wilk test is a hypothesis test that compares sample group test statistic W to a critical value of W. If test statistic W is higher than the critical value of W, the Null Hypothesis is not rejected. The Null Hypothesis of the Shapiro-Wilk normally test states that sample group is normally distributed. The following results indicate that the test statistic W for the data group of each factor level is greater than its respective critical W value. All factor levels are deemed to have normally distributed data. (Click On Image To Se a Larger Version)

The individual Shapiro-Wilk normality tests for the data groups of each level will be shown as follows. The critical W values are taken from a table based upon n (the number of data observations in the sample group) and α (the Level of Significance, set to 0.05 here).

Shapiro-Wilk Normality Test in Excel of Factor 1 Level 1 Data (Click On Image To Se a Larger Version)

Shapiro-Wilk Normality Test in Excel of Factor 1 Level 2 Data (Click On Image To Se a Larger Version)

Shapiro-Wilk Normality Test in Excel of Factor 1 Level 3 Data (Click On Image To Se a Larger Version)

Shapiro-Wilk Normality Test in Excel of Factor 2 Level 1 Data (Click On Image To Se a Larger Version)

Shapiro-Wilk Normality Test in Excel of Factor 2 Level 2 Data (Click On Image To Se a Larger Version)

Test Statistic W is larger than W Critical in all five cases. The Null Hypothesis therefore cannot be rejected. There is not enough evidence to state that any of the data groups is not normally distributed with a confidence level of 95 percent.

Correctable Reasons Why Normal Data Can Appear Non-Normal

If a normality test indicates that data are not normally-distributed, it is a good idea to do a quick evaluation of whether any of the following factors have caused normally-distributed data to appear to be non-normally-distributed:

1) Outliers

– Too many outliers can easily skew normally-distributed data. An outlier can often be removed if a specific cause of its extreme value can be identified. Some outliers are expected in normally-distributed data.

2) Data Has Been Affected by More Than One Process

– Variations to a process such as shift changes or operator changes can change the distribution of data. Multiple modal values in the data are common indicators that this might be occurring. The effects of different inputs must be identified and eliminated from the data.

3) Not Enough Data

– Normally-distributed data will often not assume the appearance of normality until at least 25 data points have been sampled.

4) Measuring Devices Have Poor Resolution

– Sometimes (but not always) this problem can be solved by using a larger sample size.

5) Data Approaching Zero or a Natural Limit

– If a large number of data values approach a limit such as zero, calculations using very small values might skew computations of important values such as the mean. A simple solution might be to raise all the values by a certain amount.

6) Only a Subset of a Process’ Output Is Being Analyzed

– If only a subset of data from an entire process is being used, a representative sample in not being collected. Normally-distributed results would not appear normally-distributed if a representative sample of the entire process is not collected.

Nonparametric Alternative For Two-Way ANOVA W/ Replication When Data Are Not Normal

When groups cannot be shown to all have normally-distributed data, a relatively unknown nonparametric test called the Scheirer-Ray-Hare Test should be performed instead of Two-Factor ANOVA With Replication. The Scheirer-Ray-Hare Test test will be performed on the original sample data in a blog article shortly following this one.

The Friedman test is occasionally mentioned as an alternative but that is incorrect. The Freidman test is a nonparametric alternative for Repeated Measure ANOVA but not for Two-Factor ANOVA With Replication.

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Statistical Topics and Articles In Each Topic

• Histograms in Excel
• Bar Chart in Excel
• Combinations & Permutations in Excel
• Normal Distribution in Excel
• t-Distribution in Excel
• Binomial Distribution in Excel
• z-Tests in Excel
• t-Tests in Excel
• Hypothesis Tests of Proportion in Excel
• Chi-Square Independence Tests in Excel
• Chi-Square Goodness-Of-Fit Tests in Excel
• F Tests in Excel
• Correlation in Excel
• Pearson Correlation in Excel
• Spearman Correlation in Excel
• Confidence Intervals in Excel
• Simple Linear Regression in Excel
• Multiple Linear Regression in Excel
• Logistic Regression in Excel
• Single-Factor ANOVA in Excel
• Two-Factor ANOVA With Replication in Excel
• Two-Factor ANOVA Without Replication in Excel
• Randomized Block Design ANOVA in Excel
• Repeated-Measures ANOVA in Excel
• ANCOVA in Excel
• Normality Testing in Excel
• Nonparametric Testing in Excel
• Post Hoc Testing in Excel
• Creating Interactive Graphs of Statistical Distributions in Excel
• Solving Problems With Other Distributions in Excel
• Optimization With Excel Solver
• Chi-Square Population Variance Test in Excel
• Analyzing Data With Pivot Tables
• SEO Functions in Excel
• Time Series Analysis in Excel
• VLOOKUP