This is the last research/statistics column (please hold the applause). I will be switching to an exciting new topic format in 2003 - so stay tuned. It is only fitting that we finish 2002 with an examination of inferential statistical formulas. They essentially measure differences between groups. Knowing the main formulas, and how one goes about choosing the best one for a particular job is the theme of this column.

We stay true to our initial claim that you can understand these things without esoteric mathematics. That can be a big step to actually taking on the math once you feel comfortable with the concepts.

The big divide
Statistical formulas are divided into two distinct camps - the parametric and non-parametric. We'll examine four formulas in parametric and two in the non-parametric. The parametric formulas (Vogt, 1999; Fraenkel & Wallen, 2000) have a number of assumptions about the population and any sample taken from that population. In other words, the data has certain characteristics. For example:

* They approximate a normal distribution.
* They can be measured with interval or ratio scales.

Examples of parametric data come from test/instrument scores and other such measures. The non-parametric formulas, as you might guess, do not make many assumptions about the population or any sample taken from that population. For example:

* The distributions are generally not normal.
* Most data are measured with nominal and ordinal scales.

Examples of non-parametric data include age, gender, or a scale that asks you to rank your favorite type of therapy.

A few key parametric formulas
There are a ton of statistical formulas. It is enough to make your head spin, unless you know that most are variants of a few standard ones. Those standard formulas are the ones we will review, and as we have done all along, we keep the explanations uncomplicated.

Let's start out with a classic parametric statistical formula. It is called the t-test for independent means (Kranzler & Moursund, 1999). This is a test to see if the difference between two sample means is significant. One use of the t-test, is to see if there is a difference between a sample of clients who are given one form of treatment, and a sample who are not. (Note how easily this would fit into a true or quasi experiment design.) The result is given as a t score. This score is checked against a statistical table to see if the score is significant. In the journals, the score is a mathematical statement that reads something like the following (Salkind, 2000):

t = 3.43, p < .05

Let's break this phrase down:
* "t" is the statistic to determine differences between groups.
* 3.43 is the figure obtained from the actual mathematics obtained from the t-test formula.
* The most important thing in this little statement is the phrase p < .05. This means that probability is less than 5 percent that 3.43 has exceeded the critical value for significance or rejection of the null hypothesis. (If 3.43 did exceed the critical value (.05) then "p" would be more than .05, i.e., .06, .07, .08, etc., and then the null hypothesis would be the best explanation.)

Depending on the published research, you sometimes will find an asterisk (*) beside numbers in a table. At the bottom of the table you will see another asterisk (*) followed by the phrase p < .05. This simply means the number marked by the asterisk was significant at the .05 level. Understand this little phrase, and you have achieved a milestone.
A slightly different t-test can be used to measure the difference between the same groups, but tested at different times.
But what if you want to find out if there is significant difference between the means of more than two group variables. In this case, you resort to a formula called a simple analysis of variance (ANOVA) (Fraenkel & Wallen, 2000).

It is a variant of the t-test, but measures significance between three or more groups on one or more variables (e.g., determining the difference in drinking rates between one group given one form of treatment, the second group given another form of treatment, and a third group serving as a control group). The ANOVA gives a score called the F ratio, and like the t-test, is compared against a standardized table. If you understood this, you now understand the basics of the ANOVA.

Next, on the basic formulas list is the analysis of covariance (ANCOVA) (Vogt, 1999; Fraenkel & Wallen, 2000). Using our example for the ANOVA, say you want to control for any extraneous variables (e.g., number of AA meetings attended per month) that you feel would interfere with the results of the three groups. Using this formula, you could remove those covariates from a list of possible explanations that would affect the dependent variable (in our case the drinking rates). The score is given in as an F ratio score similar to the ANOVA. The actual statistic is rather formidable, but this is the basic concept. Understand it, and you understand the ANCOVA.

The last parametric formula we examine is called the multivariate analysis of variance (MANOVA) (Vogt, 1999; Fraenkel & Wallen, 2000). Sounds imposing, but whereas the ANOVA only addresses one dependent variable, the MANOVA addresses more the two dependent variables in its calculations. Staying with our example, this test (also given as a F ratio) would let you calculate drinking rates and number of arrests. Again, as far as the concept, that's it. Understand this simple explanation and you got the MANOVA.

Note all parametric formulas are related to one another, except they measure slightly different outcomes. They are variants of one another. Many statistical formulas are that way.

Two non-parametric formulas
We examine two last formulas in the non-parametric realm. The first is called the Mann-Whitney U Test (Vogt, 1999; Fraenkel & Wallen, 2000). It too is an alternative of the t-test, but in this case, it measures differences in ranked data (not interval like the t-test). It produces a U score, and like all the tests we examined, (after you do the math) you compare the U score to a statistical table to determine significance.

For instance, the Mann-Whitney would investigate a hypothesis that alcohol clients who receive antagonist drug naltrexone will report fewer urges for alcohol than those who did not receive the drug. Note that this formula does not give you as much information as the parametric equivalents. These types of formulas never do, hence they are not as powerful as the parametric ones, but useful at times.

Our last non-parametric test is called chi-square (X2) (Vogt, 1999; Fraenkel & Wallen, 2000). This statistical formula analyzes data reported in categories (grouping data into social categories such as age or gender and counting the number of the examples in the category.) The test is based on the comparison between expected frequencies versus actual frequencies. If the actual measure does not different from expected measure, then we can say the groups do not differ. For example, the chi-square can answer the hypothesis that men's use of the local outpatient unit does not differ from women's use of the same center. Again, not much information to be gained, but simple and useful in some cases.
Remember that all these explanations were presented in very streamlined fashion. There is quite a bit more that is involved, but the aim was to grasp the basic concepts. Should you dare to dig deeper, just check out the references listed.

Final thoughts
Research and statistics have never been popular subjects because of their presumed difficulty, and presumed lack of relevance to actual counseling. This column has tried to convince the reader that both are a myth. This material is not difficult, and is certainly understandable when presented in an easy-going manner. Moreover, we gave ample examples of how statistics can enhance your counseling skills. Having such an understanding is extremely important, no - critical, in this day and age.

If you stuck it out and read all the columns, you have gained a solid grounding in research and statistics. That makes you more valuable to your program and clients. You can now begin to conduct your own research. Such a capability makes for almost unlimited possibilities. And, with less fear and trepidation, you can read more journal articles, and your clients will be the ones to benefit. I wish you the best.

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