Statistics - Part 3: Advanced Statistics

This is the third in a series of three articles that address the underlying principles to help you analyze your data without having to be a statistician.

The first step in any data analysis strategy is to determine what you want to know, or your purpose in analyzing the data. Ideally, you should have determined this before collecting your data, but all too frequently this is not the case. Many of the commonly used statistical tests can be classified into one of three categories:

Analyses for Description

The purpose of descriptive statistics is to describe the data. The type of data will determine which descriptive statistic is appropriate. Specifically, one can only calculate a mean with interval or ratio data, whereas a mode can be calculated with nominal, ordinal, interval or ratio data.

Analyses for Comparison

A common goal in conducting research is to determine if differences exist between two or more groups. For example, we may be interested in determining if people who defect to another service provider are different from those who choose to remain. While the most common examples of this type of analysis focus on differences of means and variances, it is important to note we can analyze many types of differences including correlation coefficients, proportions and percentages. The statistic used is determined by the type of data you have.

Nominal Data: Chi-Square

The chi-square test is used to determine if a relationship between variables exists by comparing expected and observed cell frequencies. Specifically, a chi-square test will examine the observed frequencies in a category and compare it to what would be expected by chance or if there were no relationship between the variables.

Example:

A researcher is interested in examining if peoples' preferences for winter sports are related to their preferences in automobile manufacturers. The researcher would gather data from a number of people who prefer different winter sports and automobile manufacturers and examine the relationship using a chi-square table that may resemble the following:

Auto Preference	Snow-board	Downhill Ski	X-Country Ski
X	10	25	15
Y	30	10	15
Z	5	10	15

Using the standard formula for a chi-square test, we determine that the observed frequency is indeed different form what we would expect if there were no relationship between winter sport and automobile preference (approximately 15 for each automobile preference). "Keep in mind that this only indicates a relationship exists, it does not tell us that one factor causes another.

Interval Data: t-Test

One of the most common statistical tests to use for comparisons with interval data is the t-test. The t-test compares the means of two groups, and then determines whether those two means are different enough to be statistically different.

Example:

A researcher is interested in assessing whether or not there are any differences in on-the-job-performance of employees who receive training on-site compared to employees who travel off-site for training. The researcher would collect the appropriate performance data for the two groups, subject the data to a t-test and interpret the results (the value of the t statistic is compared to a table indicating whether or not the observed means are statistically different). If the results indicate that there is no difference in the performance of the two groups of employees and the data have been collected with the appropriate amount of experimental controls, the researcher may be inclined to conclude that the two locations for training are equivalent and make a recommendation about consolidating the training.

Interval Data: One-Way ANOVA

While the t-test is useful for testing differences between two groups, frequently we are interested in more than two groups. In those cases, we often rely on the Analysis of Variance (ANOVA) To tell us if those groups are different on some variable of interest. For example, if the training example from above included a third group (i.e., a combination of on- and off-site training) it would require use of the ANOVA instead of the t-test.

Interval Data: Factorial ANOVA

Frequently we are interested in understanding the effects of varying levels of two or more variables on a third variable. In such a case, we are unable to use the One-way ANOVA because it is limited to comparisons of the effects of one variable on another. Essentially, a factorial ANOVA analyzes the impact of both the variables independently as well as jointly to determine how they affect another variable of interest.

Example:

Continuing with our training example, another factor important in determining job performance might be job performed. Specifically, it might be both variables (or factors) that are important in determining job performance with certain types of jobs responding better to on-site training and others responding better to off-site training. In order to examine whether or not this is true, the data should be analyzed using a factorial ANOVA.

In this case we might select the three types of training (on-site, off-site, both) and two job levels (e.g., entry-level, mid-level) and examine their effects on job performance both alone and in combination. These two factors yield the following hypothetical results:

Training

Job Level	On-Site	Off-Site	Both
Entry	Hi Sat.	Low Sat.	Low Sat.
Mid	Low Sat.	Hi Sat.	Low Sat.

As the table indicates, the effectiveness of the type of training depends on the level of the job. This interaction effect (depicted in the following graph)means, quite simply, that the two factors "interact" to determine the effect on the variable of interest. A common way of interpreting interaction is to use the phrase "it depends." Specifically, if someone were to ask about the relationship between training location and performance, you would have to say, "it depends upon job level." Essentially, an interaction is an indication that an observed relationship is conditional, or depends on the values of another variable.

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