Matching Statistics with the Research Design
GO TO PART ONE OF RESEARCH DESIGNS
This document focuses on how to select the correct statistical procedure for your research study. This is an important element of your research design. The choice of how the data is analyzed effects the number of subjects and the way you measure the dependent variables. To avoid using too few subjects or gathering the wrong type of data, you need to decide on the data analysis method during the design phase of your research.
A key element to selecting the correct statistical test is to fit the design one is using into one of four patterns. The patterns are made by the sequence of events; the Xs and Os of the design. You must evaluates the design diagram by looking at the pattern of Xs and Os for group 1, by counting the number of Os for group 1, and by knowing the number of groups. The simplest pattern is pattern 0. This pattern consists of at least one group and one observation and is used for descriptive and relationship research questions.
EVENT 1 ------------------- GROUP 1 O (Pattern 0) -------------------
The remaining three patterns are used for differences research questions. Pattern 1 is an experimental design that is all observations and has one or more groups.
EVENT 1 ------------------- GROUP 1 O GROUP 2 O (Pattern 1) GROUP 3 O ------------------- EVENT 1 2 ----------------------- (Pattern 1) GROUP 1 O O -----------------------
Pattern 2 is an experimental design that has a pattern of events where first a treatment is done and then the DV is observed. The DV may be observed more than once and still the design is assigned to pattern 2.
EVENT 1 2 3 -------------------------- GROUP 1 X1 O1 O2 (Pattern 2) GROUP 2 X2 O1 O2 --------------------------
Pattern 3 is for experimental designs that have the sequence of events: observe, treat, observe. It is usually assumed that the first observation is used as a control measure. Pattern 3 designs are assigned to analysis of covariance when the DV uses interval measurement and there are three or more observations of one group, or two or more observations of two or more groups. In this case, the first (pre-treatment) observation is used to predict the post treatment observations and then the residuals are analyzed for differences due to the treatment.
EVENT 1 2 3 ----------------------------- GROUP 1 O1 X1 O2 GROUP 2 O1 X2 O2 (Pattern 3) GROUP 3 O1 O2 -----------------------------
The blank cell for group 3, event 2 indicates that group 3 is a control or no treatment group. Designs can have any number of events, but you only need to count the total number of events and observations, and examine the first three events, to deduce the design's pattern.
Once you have the pattern you can use Table 2 to locate the correct statistical test.
TABLE 2 Relating Research Designs to Appropriate Statistical Analyses ------------------------------------------------------------------- DESIGN STATISTICAL TEST ------------------------------------------------------------------- DIFFERENCES RESEARCH QUESTION 1. Basic two-group design 1. a. t-test - independent means (Pattern 1 or 2) (Interval or ratio data)* b. Mann-Whitney U test (Ordinal data) c. Chi-square (nominal data) 2. Pre-test and post-test 2. a. t-test - dependent design. (Pattrn 3) (non-independent) means (Interval) b. Wilcoxon or Sign test (Ordinal) c. McNemar test (Nominal) 3. Time-Series or Single 3. Interrupted time-series analysis Subject (Pattern 3) (interval) 4. Covariance, or repeated 4. a. Repeated measures analysis measures design. of variance OR Analysis of (Pattern 1 or 3) co-variance (Interval) b. Friedman's AOV by ranks (Ordinal) c. Cochran's Q (Nominal) 5. Three or more groups 5. a. Analysis of variance design (Pattern 1,2 or 3) (Interval) b. Kruskal-Wallis AOV (Ordinal) c. Chi-square test for K independent groups (Nominal) DESCRIPTIVE RESEARCH QUESTION (Pattern 0) 6. One-group sample from a 6. a. One-group t-test (Interval) known population. b. Kolmogorov-Smirnov test for goodness-of-fit (Ordinal) c. Chi-square goodness-of-fit test (Nominal) RELATIONSHIPS RESEARCH QUESTION (Pattern 0) 7. Correlational study 7. a. Pearson product moment (Two or more variables correlation coefficient. and one group) (Interval) b. Spearman's rank order correlation, Kendall's Tau (Ordinal) c. Lambda Beta, Phi coefficient or Chi-square (Nominal) ___________________________________________________________________ * Refers to the way the dependent variable is measured. Continuous variables use interval or ratio measurement Categorical variables use nominal or ordinal measurement
For example, a one group pre-post design (Example 1), with interval measurement of the DV, will need the t-test for dependent means.
EVENT 1 2 3 -------------------------- (Example 1) GROUP 1 O1 X O2 --------------------------
If you changed the scale of measurement used by the DV to ordinal, then you would use the Wilcoxon or Sign test instead of the t-test.
Another example. Given the two-group design shown below.
Type of research question: Differences Number of independent variables: 1 Type of variable : categorical Number of dependent variables : 1 Type of variable : continuous Number of Events : 3 Number of Groups : 2 EVENT 1 2 3 ------------------------ GROUP 1 O1 O2 O3 GROUP 2 O1 O2 O3 ------------------------ Statistical test to use: Use a one factor repeated measures analysis of variance.
This section describes the uses for the statistical tests that are available to the researcher for data analysis. Table 2 provides a summary of this section.
Categorical IV and Continuous DV
All the tests in this section are for differences research questions. These tests are appropriate for large sample sizes, i.e., twenty or more subjects per group. Some of them can be used with smaller group sizes (10 to 15 subjects per group), but the validity of the test results can be reduced when a small group size is used.
1. t-test for independent means - Use this test when you have two-groups and a single measurement of the dependent variable (DV). This test tells you if the mean of one group is different from the mean of the other. Remember all continuous variables use interval or ratio measurement.
2. t-test for dependent means - Used when you have one group and you measure the DV twice. It tells you if the mean of the first measurement is different from the mean of the second.
3. Analysis of variance (AOV) - Used when you have three or more groups and measure the DV once. It tells you if one of the group means is different from the other means. If you have one independent variable (IV) it is a one-factor AOV, and if you have two IVs it is a two-factor AOV. The two-factor AOV tells you three things: 1) if there is a difference between the DV means for the first IV, 2) if there is a difference between the DV means for the second IV, and 3) if there is a difference between the group means (An interaction effect; see Chapter 4 for more details). An AOV design with two factors has one group for each combination of the categories of the two IVs. If one IV has 2 categories and the second IV has 3 categories then the study will have 6 groups (3 x 2). AOV designs (and RAOV and AOCovar below) can be used with as small as four subjects per group and still obtain valid results.
4. Repeated measures analysis of variance (RAOV) - Used when you have three or more groups and measure the DV two or more times. Can have one or more IVs. In addition to the information provided by an AOV design (see item 3 above) it tells you if the groups are different across the several measurements of the DV. Also appropriate for three or more observations of one group.
5. Analysis of covariance (AOCovar) - Used when you measure the DV before and after the treatment, have two or more groups, and have one or more after treatment observations. You can have one or more IVs. The t-test for dependent means is the one group, two observations equivalent of the AOCovar. Analysis of covariance is most useful when the groups are not randomly assigned and you want to control for any initial differences between the groups.
6. Multivariate analyses - When you have two or more DVs and three or more groups you can also run a multivariate AOV, RAOV, or AOCovar. This test allows you to see how the DVs relate to each other. It tells you if the DVs overlap in terms of what they measure or if they each measure a separate trait or concept. For example, you may use two different measures for heart rate (resting and immediately after exercise) as DVs and your single IV may have two categories of exercise levels. A multivariate AOV will tell you about the effects of the exercise levels on each heart rate measurement. Your results will show that the resting heart rate and heart rate immediately after exercise are correlated (e.g., they are both measuring the same trait) but that the immediate measurement is effected more by exercise level while resting heart rate is effected more by overall physical condition. If you have only two groups you use Hotelling's T-squared statistic.
7. One group t-test - If you only have one group, a descriptive research question, and a single measure of the DV you can see if your sample mean is different from the mean of the population (that is you can if you know the population mean). Since you have only one group you do NOT have an independent variable. This test is useful to see if your sample is representative of the population.
Categorical IV and DV
The tests in this section are called non-parametric tests because they are based on categorical data. They can be used with small sample sizes between five and twenty per group.
1. When data are frequencies (counts of the number of times an observation falls into one of several nominal categories) use these methods.
a. Chi-square - There are many uses for Chi-square tests, here we will discuss three uses:
i) When you have one set of observations, one DV, and one group you can use the Chi-square goodness-of-fit test to see if your sample frequencies are the same as the population frequencies. This is similar to the one-group t-test.
ii) When you have one set of observations and one or more groups, one or more IVs, and one or more DVs you can see if a relationship exists between the variables or the groups. Similar to a correlation but it only tells you if a relationship exists not the strength of the relationship (as does a correlation coefficient).
iii) You can also use Chi-square for the same purpose as a t-test for independent means or a one-way AOV when you only have frequency data. A Chi-square test can be used for both a relationship question and a differences question because at this level both questions have the same answer. If a relationship exists then the groups are different. If there is no relationship then the groups are the same. If there is a relationship, then you must use your knowledge of group membership to interpret the results. The frequencies in the categories of the DV are different is a systematic way if a relationship exists.
b. Cochran's Q - Used when you have three or more observations of a nominal variable and one group. This test is the non-parametric parallel to the repeated measures AOV.
c. McNemar test - Used with two observations of one group; nominal data. Non-parametric equivalent of the t-test for dependent means.
d. Phi coefficient - This is a measure of relationship used when you have two variables that each only have two categories (e.g., yes/no or 1 / 0). This type of variable is called a dichotomous variable. This measure gives the same result as Pearson's product moment correlation coefficient.
e. Lambda beta - This is a measure of relationship used when you have two nominal variables that each have two or more categories. Lambda beta is a more general measure of association than the Phi coefficient. This measure is not the same as Person's product moment correlation coefficient.
2. When data are scores (ordinal measurement) use these methods. These can be used with interval data as well by converting the interval data to ranks.
a. Spearman's rank order correlation - This is a measure of relationship (association) used when you have two ordinal (rank order) variables. It is the non-parametric parallel to Pearson's product moment correlation coefficient.
b. Kendall's Tau - This method provides the same measure of relationship as the Spearman. It is best to use Kendall's method when you have ten or fewer subjects.
c. Kolmogorov-Smirnov test for goodness-of-fit - Used to compare your sample distribution to the population distribution. Use this test when you have ordinal (rank order) data. This test provides the same information as the one-group t-test or Chi-square goodness-of-fit test.
d. Mann-Whitney U-test - Used when you have two groups and observe the DV once; ordinal data. This test is the non-parametric equivalent of the t-test for independent means.
e. Wilcoxon Matched-pairs, signed-ranks test - Used when you have two observations of one group; ordinal data. This test uses the ranks as data. It is similar in function to the t-test for dependent means and the McNemar test for nominal data.
f. Sign test - Used for exactly the same data as the Wilcoxon Matched-pairs, signed-ranks test. This test is not as powerful as the Wilcoxon because it only uses direction of change (+ or -) and not the actual ranks.
g. Kruskal-Wallis one-way analysis of variance by ranks - Used when you have three or more groups and one observation of an ordinal variable. This test is the non-parametric parallel to the AOV.
h. Friedman analysis of variance by ranks - Used when you have three or more observations of one ordinal DV and one group. It provides the same information as Cochran's Q or the Repeated measures AOV except that it is used with rank order data. This test is sometimes called a two-way AOV because the subjects are considered as the second factor. It is not a true two factor design.
Continuous IV and DV
All of these procedures are used with relationships research questions and require large sample sizes, i.e., twenty or more per group. Smaller sample sizes can be used (10 to 15 per group) but the validity of the results can be reduced.
1. Pearson's product moment correlation coefficient - Used with pairs of continuous variables. It provides a measure of the degree of relationship between the two variables.
2. Regression analysis - Also called univariate regression analysis because there is only one DV used. This is a mathematical procedure not a statistical test. It produces an equation that lets you predict an individual's score on the DV from his score on the IV. The equation looks like this:
DV = I + IV(S)
Where I is called the intercept and S is called the slope. The regression equation does not tell you the accuracy of your prediction. The correlation coefficient tells you the accuracy of the prediction.
3. Multiple regression analysis - This procedure combines the functions of Pearson's correlation and regression analysis when you use two or more IVs to predict one DV. The procedure provides a regression (prediction) equation that looks like this:
DV = I + (IV1 * B1) + (IV2 * B2) + (IVn * Bn)
Where I is the intercept, and B1, B2, and Bn are called beta weights (Bn is the nth beta weight for the nth IV). The beta weights are values that correspond in function to the slope used in the univariate regression analysis. Multiple regression analysis also provides a measure of relationship called Multiple R. This Multiple R (or R squared) tells you the accuracy of prediction of the DV from the several IVs.
4. Multivariate Multiple Regression Analysis - This procedure is used when you have several IVs (like multiple regression above) and you have several DVs. This procedure provides a set of prediction equations (one for each DV), tells you the accuracy of prediction of each DV from the IVs, and finally tells you the degree to which the DVs are related and lets you study the structure of the relationship.
There are no statistical tests for designs with continuous IVs and categorical DVs. Relationship designs require that the IVs and DVs both use the same type of measurement, i.e., both categorical, or both continuous.
The computational procedures for these statistical procedures can be found in many books on statistics. Three good books for this purpose are: Statistical Principles in Experimental Design by B. J. Winer; Non-parametric Methods of Quantitative Analysis by Jean D. Gibbons; and Statistical Methods in Education and Psychology by Gene V. Glass and Julian C. Stanley.
There are several computer programs available that ease the pain of computing statistics by hand. Three major ones are: SPSS-Statistical Package for the Social Sciences by Norman H. Nie, C Hadlai Hull, Jean G. Jenkins, Karen Steinbrenner, and Dale H. Bent; SAS-Statistical Analysis System from SAS Institute Inc.; and BMD-Biomedical Computer Programs by W. J. Dixon.
Some on-line statistical programs are available from the US Center for Disease Control. Epi Info, a statistics package which is available for free from the CDC web site. You can read about it here - http://www.cdc.gov/epiinfo.
Two excellent books that combine statistical procedures with computer programs you can use to compute statistical tests is Multivariate Data Analysis by William W. Cooley and Paul R. Lohnes and Computational Handbook of Statistics by James L. Bruning and B. L. Kintz