Anova-Ind

Evaluating the PCH of Chance for the effect of one IV on a DV. As we pointed out at the beginning of the Interaction lecture, it is natural to start out with simple questions in experimental research: Does an IV cause changes in a DV? Does Psychotherapy improve mental health? Does a new keyboard increase typing speed? Does imagining perfect runs down a ski slope decrease the elapsed time of elite skiers? The t for independent means and the t for correlated means along with 1-Way ANOVA for independent groups and 1-Way ANOVA for correlated groups are excellent inferential statistics for evaluating the PCH of Chance for such questions as those. In 1-Way ANOVA the "1-Way" refers to the fact that we are only analyzing the effects of one IV on a DV.

Evaluating the PCH of Chance for the effect of two IV's on a D. Frequently we have more than one IV whose effects on a DV interest us. Often we want to study the effect of two IV's on one DV in the same experiment. We will now consider another powerful inferential statistic--the 2-Way ANOVA for independent groups. The 2-Way ANOVA will allows us to evaluate the PCH of Chance when we are examining the effects of two IV's on a DV. The "2-Way" in 2-Way ANOVA refers to the fact that we are analyzing the effects of two IV's on a DV.

To understand the 2-Way ANOVA, it is necessary to understand the idea of interaction.

Consequently, this lecture continues our discussion of the concept of interaction that was started in the Interactions lecture. You may want to review that lecture before continuing with this lecture. As a very short review, the slide below gives the the verbal definition of an interaction between the effects of two IV's.

Three kinds of effects. Before we can talk about the F-tests involved in the 2-Way ANOVA, we have to define three kinds of effects: The Main Effect of IV1, the Main Effect of IV2, and the Interaction Effect. We have developed the idea of Interaction Effects in the Interaction lecture. But we have not yet talked about Main effects.

Pollution Example. Lets go back to the pollution example from the Interaction lecture so that we have some concrete numbers to work with in defining these three kinds of effects. The dependent variable is a measure of health, on some kind of scale from 0 to 100, where 100 is perfect health, and 0 is mortally critical illness. IV1 is the presence or absence of sulfur dioxide and IV2 is the presence or absence of carbon monoxide in the air. As we mentioned before, the data have been sketched out solely for illustrative purposes.

Experimental Design of the Study. Using those two independent variables we designed a hypothetical four-group study. One group (upper cell on the left in the table of means)) breathes clear air, neither carbon monoxide (CO) and nor sulfur dioxide. A second group (upper right hand cell) breathes no carbon monoxide, but does breath sulfur dioxide. A third group (in the lower left cell), breathes carbon monoxide but no sulfur dioxide. The fourth group (lower right cell) breathes both gases.

If you had 100 participants you would randomly divide them up into 25 participants in each of those groups, or if you have 40 subjects, you would randomly break them up into 10 in each of those groups. That's the experimental design of the study.

Cell Means. The the cell means in the table are based on hypothetical data. We have a control group with clean air, whose mean health score is 96. We have a group that breaths only sulfur dioxide whose mean health score is 80. We have a group that breaths only carbon monoxide whose health score is 68. And we have a group that breaths both whose health score is 8.

Column Means. The top of the left-most column says "No SO2." Notice that we can average the two groups that received No SO2. Their cells means are 96 and 68 which averages to 82. So the average of all participants (irrespective of CO level) who breathed No SO2 is 82. The top of the right-most column says "SO2." All participants in that column did breathe SO2. We get the mean of all these SO2-breathers by averaging 80 and 8, which is 44.

Examine the table of means to be sure you understand how we are averaging the cell means to get column means. The column means disregard the effects of CO by averaging across the levels of CO.

Row Means. Look at the Table of Means again. You will see that we can also get row means. The top row is named "No CO." The top row of cell means, 96 and 80, which average to 88. That means, disregarding SO2, the average of all participants who did not breath CO is 88. Now let's look at the bottom row (participants who did breathe CO). The cell means are 68 and 8 which averages to 38.

Main Effect of SO2. The current graphic shows the table of means along with a graph of the main effect of SO2. The main effect of SO2 is found by examining the difference between the two column means in the table. The mean for the "No SO2" column is 82, while the mean for the "SO2" column is 44. On the average, people NOT breathing SO2 had a health rating of 82, while does who did breathe SO2 had a health rating of 44. The difference between these means (82 minus 44 = 38) ) gives us the main effect of breathing SO2. In this study those who breathed SO2 were rated 38 points lower in health than those who did not breathe SO2.

Notice that the column means are averaged over the levels of CO. Among those people in the No SO2 column, are some who breathed CO and some who did not breathe CO. So what the jargon "main effect of SO2" refers to is the effect of SO2 averaged over levels of CO.

Main Effect of CO. In a corresponding way, the main effect of CO is found by examining the difference between the two row means. The mean for the "No CO" row is 88, while the mean for the "CO" row is 38. The difference between these means (88 minus 38 = 50) ) gives us the main effect of breathing CO. In this study those who breathed CO were rated 50 points lower in health than those who did not breathe CO.

Notice that the row means are averaged over the levels of SO2. Among those people in the No CO column, are some who breathed SO2 and some who did not breathe SO2. So, as above, what the jargon "main effect of CO" refers to is the effect of CO averaged over levels of SO2.

The next graph shows the main effect of CO visually. Make sure the connection between the table of means and graph is clear to you. An important and useful skill is integrating numerical and graphical information into a solid understanding.

INTERACTION EFFECTS . The current graphic shows the table of means along with a graph of the interaction effect. We have already talked about this interaction pattern in the Interaction Lecture. You can infer the interaction of SO2 and CO from the pattern of cell means in the table. The graph shows this pattern visually.

A free choice. When you graph an interaction effect you are free to choose which IV you put along the horizontal axis. I have put the two levels of the SO2 variable along the the horizontal axis. Then I drew a separate line for each level of the CO variable. I could just as well have put the CO variable along the horizontal axis and indicated levels of of SO2 by separate lines. You usually make the choice of which variable to put along the horizontal axis based on how well the resulting graph illustrates the conceptual point you are making.

Putting the cell means on a graph. I put the two levels SO2 (no SO2 and some SO2) along the horizontal axis and indicated the levels of CO with two lines, one for no CO and one for the presence of CO. Looking at the two lines we can see the means of our four groups. The mean health rating of 96 is for the the group who breaths clear air, neither carbon monoxide or sulfur dioxide. The mean health rating of 80 along the same (No CO) line is for the group who's breathing sulfur dioxide, but is not breathing any carbon monoxide (this is the group who is breathing sulfur dioxide alone). If we drop down to the leftmost point on the lower line the mean health rating of 68 is for the group that has no sulfur dioxide but it is breathing carbon monoxide. Finally the mean health rating of 08 is for the group that breaths both of the gases.

Interaction pattern. Two IV's interact if the effect of one of them depends on the level of the other. In this hypothetical data we can see that there's an interaction, that both of the gases produce a deleterious effect on health, but that the negative effects on health are especially strong when they're combined. Without CO (top line on the graph) the effect of the presence of SO2 is a loss of 16 health rating points (96 - 80 = 16). But when CO is present (bottom line) the effect of the presence of SO2 is a loss of 60 health rating points (68 - 8 = 60). In other words the size of the effect of breathing SO2 depends on whether CO is present or not. There a synergy here in which the combination of the two gases does much more damage than would be thought by simply adding up the effects of the individual gases.

In Summary. The difference between row means reflect the main effect of one of the two IV's, the difference between column means reflects the main effect of the other IV, and the pattern of differences among the cell means reflects the interaction effect.

Most research on environmental pollutants is done looking at the effects of substances alone in isolation from other substances. It is important to know that these substances might interact with other substances to produce interaction effects beyond the main effects of each substance.

We will now go on to the therapy example that we started in the interaction lecture. I want to give you a rich set of examples to think about because my experience is that the distinctions among interaction effects and main effects are usually something that people haven't thought about prior to taking a statistics class. Yet they are very important critical thinking issues independent of statistics. It is important to be able to untangle main effects and interaction effects when you are understanding conversations about research results. And what requires even more practice for beginning students learning how to look at table of data and naturally find the main and interaction effects and visualize them on a graph.

Type of Therapy and Presenting Problem. In our example the Type of Therapy IV has two levels--cognitive versus behavioral. The Presenting Problem IV has two levels--learning problems and habit problems (like chewing fingernails). Again we have a four-group study. One group consists of people who are having trouble learning and receive cognitive therapy; one group has learning difficulties and receives behavior therapy; one group has a habit they want to break and receives cognitive therapy and a fourth group has a habit they want to break and receives behavior therapy.

The data in the table are made up and highly schematized to illustrate a classic case that has an interaction effect and no main effects.

Main effect of Type of Therapy. The difference between column means, averaging across cells, yields the main effect of Type of Therapy. In this case the two column means are identical, so there is no difference between them and therefore no main effect of Type of Therapy. On the average the success of the two types of therapy came out to be the same, whether they received cognitive therapy or behavioral therapy.

Main effect of Presenting Problem. Which presenting problem is easier to fix? Are therapists more successful with solving learning difficulties or with breaking habits? The difference between the row means indicates the main effect of Presenting Problem. Row mean for learning problems is 50; so is the row mean for habits. Since there is no difference between row means, there is no main effect of Presenting Problem.

There are no main effects for either IV. From that alone it appear nothing's going on in this study--the two therapies were the same, the two presenting problems were the same. Of course something is going on, and all the action is happening in the interaction.

Classic X-shaped interaction. An interaction (or lack of it) is found in the pattern of cell means. When we graph the cell means for the Therapy by Presenting Problem example we see an X-shaped pattern. The line for learning difficulties starts high at 75 for cognitive therapy and drops down to 25 for behavior therapy. The line for habit problems is just the opposite; it starts low at 25 for cognitive therapy and rises up to 75 for behavior therapy. The two lines cross to form an X.

So cognitive therapy does well with learning problems, but when we switch to habit problems, cognitive therapy does relatively poorly. On the other hand, behavioral therapy does very well with breaking habits, but is less useful for learning problems. Which therapy is better depends on what kind of problem is being addressed. This fits with our verbal definition of an interaction. The effect of Type of Therapy depends on which level of Presenting Problem we are talking about.

This X-shaped interaction cancels out the main effects of the two IV's. In the discussion above we found that there were no main effects of either of the two IV's. But notice how both Type of Therapy and Presenting Problem really do matter. In these simulated data, cognitive therapy is much better for learning difficulties, and behavior therapy is much better for breaking habits. So Type of Therapy actually does have an effect--it's just that that effect cancels itself out across levels of Presenting Problem.

In general, it is possible that an interaction may obscure the main effects of one or more IV's.

Illumination Example. It is well known that the color of objects affects how easy it is to see them. A scientist wants to change the color of fire engines from red to greenish yellow. He thinks that red is easier to see in bright light, but that as the light gets dimmer the greenish-yellow will be easier to see than the red.

So he runs a study in a lab in which people fixate on the center of a screen and blotches of color are projected onto random locations on the screen for a short period of time. Each person is to say when he or she detects a color splotch and to name the color. The experimenter has two independent variables: The color of the splotch (red or greenish-yellow) and the illumination level (bright as sunlight, bright as overcast day, and bright as night in a city). The dependent variable is the number of times a person detects a color splotch out the twenty times that it is shown.

The resulting experimental design has six groups, one for each combination of 2 colors and 3 illuminations. Sometimes scientists use descriptive jargon like saying this is " 2 by 3 study," indicating that one IV has two levels and the other has three levels.

Variables. The dependent variable is the number of detections --we're counting the number of times an object is detected. The independent variables are color and illumination. The levels of color are red and greenish yellow, and the levels of illumination are sunlight, overcast, and city night.

Interaction Hypothesis. Two independent variables, IV1 and IV2 are said to interact if the effect of IV1 on the DV depends on the level of IV2. If there is an interaction then the detectability of colors will be different at different levels of illumination. If there is no interaction then the effect of color on detectability will not differ at different levels of illumination.

The way that the scientist stated the hypothesis we would say that he is expecting an interaction between color and delectability He expects red to be more delectable than greenish-yellow in daylight (sunlight and overcast) conditions but less delectable at night.

Graph of possible results. On the top graph we see the non-parallel lines that are the hallmark of an interaction. Red is slightly superior to green-yellow for detectability for sunlight and overcast conditions. But at night there's a reversal, red is less detectable than green-yellow. The effect of color depends on which illumination you're talking about.

On the bottom graph I've illustrated no interaction. Notice the two parallel lines. You can see that the superiority of red is exactly the same size at every level of illumination, so the effect of color is exactly the same at every level of illumination.

These graphs illustrate two possible ways the results could come out. The scientist has to run the study to find out how the data actually come out.

Three Kinds of Effects & Three Classes of Hypotheses. As we've discussed above, there are potentially three kinds of effects in a study like this--the main effect of color, the main effect of illumination and the interaction effect. Correspondingly, there will be three potential classes of scientific hypothesis--the hypothesis that color will have an effect, the hypothesis that illumination will have an effect and the hypothesis that there will be a color by illumination interaction.

Depending on the theoretical context and the particular variables a scientist may be interested in one, two or all three of these hypotheses. The way we've described the story, the scientist is primarily interested only in the interaction effect. We would say that his hypothesis (that red will be more detectable in daylight and greenish-yellow more delectable night) is an interaction hypothesis.

What's interesting about the two way ANOVA is that it allows us to evaluate the significance of all three kinds of effects

Practice. Look at the two graphs again. Think about the main effect of illumination. Do you think that illumination will have a main effect in one or the other or both graphs? Think about color, will it have a main effect on one or the other or both graphs? Think about interaction. One graph obviously shows an interaction.

However the results come out, chance will offer its pervasive question, "Did those data points come out in that particular pattern by chance alone?" One way to deal with that issue is to perform some kind of statistical analysis--in this case a 2-way ANOVA. "2-way" just refers to the fact that there are two IV's. In contrast, a "1-way ANOVA" is appropriate for a study that manipulates just one IV.

Three F Tests. Up to this point in this lecture, we've focused on the logic of main and interaction effects and not talked about statistical conclusion validity, the PCH of Chance or statistical significance. We will now develop our example more fully, examining tables of means for main and interaction effects. Then we will talk about three F tests, one for each potential effect.

An Interaction Data Pattern. The current graphic shows the table to means that goes with the Illumination by Color interaction graph we have been looking at.

Color. If you look at the table you can see that the two row means are both 81--no difference between row means. That means that there is no main effect of Color.

Illumination. The column means, in contrast, are 95, 92, and 56; these differences reflect a possible main effect of Illumination.

Interaction. The pattern of cell means as shown in the graph indicate the possibility of an interaction between Illumination and Color. When we go through the cell means row by row we see that for the top row (red), we can see that detectability starts very high, 97 in sunlight and 94 in overcast, and drops substantially down to 52 in the city night. In contrast, if we look at the bottom row (greenish-yellow) it's somewhat below red in daylight conditions, 93 for sunlight and 90 for overcast, but it doesn't drop as much at night (60). And so the effect of Illumination depends on what color you're talking about--the drop for delectability at night is more pronounced for red than it is for greenish-yellow.

Data Pattern with No Interaction. Let's examine the table of means in a case where there is no interaction between Illumination and Color. This would be how the data might turn out if red was always easiest to see, day or night.

Color. You can see the possible main effect of color reflected in the row means, 85 for red and 81 for greenish-yellow.

Illumination. There is also a possible main effect of Illumination in the column means (95, 92, and 62).

No Interaction. As I've constructed the data for this graphic there is no interaction. You'll notice that if we look at the top row in the table, the red row, it goes from 97 to 94 which is a drop of 3, and from 94 to 64 which is a drop of 30. If we look at the bottom row, the greenish-yellow row, we go from 93 to 90 which is a drop of 3--exactly the same drop as for red. And when we go from overcast to night we go from 90 to 60, a drop of 30--again exactly the same as for red. So the effect of Illumination is the same at both levels of Color. The effect of Illumination does not depend on Color--no interaction.

Two Possible Data Patterns. We've developed the Illumination and Color example in some detail. We've offered two possible ways the data might come out as a way to practice thinking main and interaction effects and as a way to practice thinking about how tables of means go with graphs. We are now going to address the issue of whether whatever data pattern does occur occurs by chance alone.

ANOVA SUMMARY TABLE. Let's look at the ANOVA source table. We won't do any calculations-- you won't have to calculate a 2-way ANOVA's, not for this course anyway. Our focus is to begin to understand the idea of multiple independent variables and interaction between these variables. You will use a computer program like StatTool to analyze homework data. There won't be any calculations, but you will need to be able to read a analysis of variance summary table and understand what's going on.

Sources of Variance. The Total Variance in the data is broken down (analyzed) into variance due to Color, variance due Illumination, and variance due to the Color by Illumination interaction and variance due to Error (variance within the 6 groups, that is variance within the cells). In ANOVA summary tables the interaction is typically symbolized by the first letters of the two variables, in this case as "C by I" or "CxI."

Error Variance. The Error source of variance is exactly the same idea as the within group term from the one way analysis of variance--now it is just the variance within the cells. In this particular example, suppose we have 10 people in each of the 6 groups. Every possible combination of color and illumination, 2 colors with 3 illuminations yields 6 groups. The 10 people in each group are all treated identically, whatever their combination of the color and illumination might be. So the variability of the detectability scores within each cell is considered to be nothing but error. This is the same logic as the within group variance in a One-way ANOVA.

Degrees of Freedom. We will have a whole section of lecture further along on how to calculate the degrees of freedom. For now note that the summary table output of any computer program will have a column for degrees of freedom. Note also that, at 10 people per cell, there is a total of 60 people in the study. This leads to the total degrees of freedom in this example being 60 - 1.

Sums of Squares. A computer program output will also have a column for SS. As I said, you won't have to calculate these, but here are the sum of squares the example data. The Total sum of squares was 1, 796.1. The sum of squares for Color was 00.0; the sum of squares for Illumination was 372.1; the sum of squares for the C by I interaction was 202.6, and the sum of squares for Error, or within cells, was 1,221.4.

Why is the SS for Color = 0? Go back and examine the table of means for the example data where there is an interaction. I'm using that table for this ANOVA. In that table the two row means are both 81. That is the mean for red = 81 and the mean for greenish-yellow is 81. Think about the concept of variance. There is 0 variance in the two numbers 81 and 81. Such a result is very unlikely to happen with real data. For this example, I have made up data that communicates as simply as possible the conceptual point that interactions can sometimes mask main effects. In doing so I arbitrarily set the mean for red overall and the mean for greenish-yellow overall equal to each other (both = 81).

Mean Squares. The mean squares are the sums of squares divided by the appropriate degrees of freedom. In our example the mean squares are 0.00 for Color, 186.01 for Illumination, 101.30 for the Cxi interaction, and 22.62 for Error.

Three F Ratios. There's going to be three F tests, one for each main effect and one for the interaction. The F ratios are simply the mean square for a source of variance of interest divided by the mean square for Error. You use the same F tables in the 2-way ANOVA as you've been using in the 1-way ANOVA.

Illumination. Let's start with the main effect of Illumination first and then return to the main effect of color. The F for Illumination is a ratio that has the MS Illumination on top (numerator) and the MS Error on the bottom (denominator). In this case F = 186.01 divided by 22.62. = 8.23.

As usual, the critical F requires two degrees of freedom--one for the numerator (top) and one for the denominator (bottom). For the numerator, the ANOVA summary table shows the degrees of freedom for Illumination (df = 2). For the denominator, the summary table shows that the degrees of freedom for Error = 54. With alpha set to .05 and degrees of freedom = 2 and 54, the critical value of F for rejecting H0 is 3.15. The obtained F ratio is 8.23, so we can reject H0. Chance is no longer a plausible explanation of why the column means differ from each other.

Color. The F test for color came out to be 0.00 (0 divided by 22.62). With an F = 0 we really don't have to bother looking up the critical value, we know that the F for Color was not significant (i.e., we could not reject H0). I've indicated this lack of significance by an "ns" for nonsignificant in the final column of the table. In this example we cannot reject the H0 and therefore we cannot reject the idea that the row means (red versus greenish-yellow) differ by chance alone.

C x I Interaction. The interaction F ratio is 101.30 divided by 22.62 = 4.47. The degrees of freedom for the interaction are 2 and 54. So the critical value of F is 3.15, just as it was for Illumination. The F for the interaction is larger than the critical value, so we can reject H0. It does not seem plausible that the pattern of cell means occurred by chance alone.

Graphical Data pattern. Previously, we considered two hypothetical graphs showing two possible data patterns. Which of these graphical patterns goes with the pattern of significance?

ANOVA Significance Pattern. Just now we presented a hypothetical ANOVA summary table with the following signficance pattern:
the main effect of color was not significant while
the main effect of illumination was significant and
the interaction effect was significant.

One graph (see illustration) showed an interaction and the other did not show an interaction. Since the ANOVA indicated that there is a significant interaction between color and illumination on detection, the ANOVA results go with the graph that shows an interaction pattern.

It is important to learn to think fluidly back and forth from a graphical data pattern to the pattern of significance. In another part of this lecture we will practice correlating the graphical representation of the data with the pattern of significance.

Summary of significance pattern . Color was not significant, illumination was significant at the .005 level, and the interaction was significant at the .025 level. That fits with the graph.

Is Color really not important? The ANOVA indicated that the effect of color was NOT significant. BUT the ANOVA also indicated that Color interacted significantly with Illumination. Those two statements contradict each other logically.

The significant interaction implies that Color has an effect that changes across levels of Illumination (If you look at the graph, you can see that red is superior in the daylight but inferior at night. So the effect of color depends on illumination.)

Logically there is a contradiction. How can Color have an effect that changes when it has no effect?

Resolution. Took at the table of means. The main effect of Color was nonsignificant in the ANOVA because the effects of Color in daylight CANCEL out the effects of color at night. Only when it is averaged over all levels of Illumination does Color have no effect. If we examine the data at each level of Illumination, Color has some kind of effect at different levels of Illumination.

You can argue logically that if an interaction is significant, then really both of the independent variables involved in the interaction are significant somewhere in the data because you can't have a significant interaction if a variable is completely ineffective. (Because if there was zero effect of Color at all levels, then it wouldn't change at different levels of illumination as the interaction implies).

Degrees of Freedom. Let's set up some standard symbols we can use. Little n is the number of participants per group. (We're going to assume the number of people are the same in every group so we can keep our formulas as simple as possible. But in general, it is possible to have different numbers in the different groups.) So let independent variable #1 be called A, and let J equal the number of levels of A. Let independent variable #2 be called B, and let K be equal to the number of levels of B. In the Color and Illumination example, J = 2 and K = 3.

Degrees of Freedom for Main Effect of A. The degrees of freedom for A are J - 1, that is, the number of levels of A minus 1.

Degrees of Freedom for the Main Effect of B. The degrees of freedom for B is K - 1.

Degrees of Freedom for the Interaction Effect. The interaction degrees of freedom are (J - 1) times (k - 1).

Degrees of Freedom for Error. The error term (denominator of F ratio) has JxK(n - 1) degrees of freedom.

Finding Critical Values. One thing you will you have to do is to look up critical F values for a 2-Way ANOVA. To look up critical values you must use the degrees of freedom we just defined. To find the critical F for the main effect of A, use J - 1 for the numerator and JK(n - 1) for the denominator of the F ratio.

To find the critical F for the main effect of B, use K - 1 for the numerator and JK(n - 1) for the denominator.

To find the critical F for the interaction effect of B, use (J - 1)(K - 1) for the numerator and JK(n - 1) for the denominator.

These are not difficult formulas, but you're going to need to practice a bit to get fluent with them. The main thing is there's three kinds of effects, and there will be three F tests. So you will have to look up three critical F values and decide, one by one, if the the F tests are signficant.

Patterns of results. Whether you're a researcher or simply a consumer of research results (reading textbooks, reading technical journals, reading popular magazines, listening to the news) at times you're likely to have to think about complex patterns of scientific findings involving more than one IV and involving distincions between main effects and interaction effects. So we are now going to practice thinking about such data patterns.

Fluid integration of significance pattersn with tables and graphs. Three common ways to represent the pattern of results in research are 1) in a table, 2) in a graph, and 3) in terms what effects are significant. What we are going to do now is to practice integrating these three ways to summarize results. If you have a table of means what's the graph of those means look like and how are the table of means and the graph related to which effects are significant. You should be able to cross-relate tables of data, graphs of data, and patterns of significance.

A, B, and AB. Just to have a short name, lets call IV1 "A". Let's call IV2 "B", and let's call the interaction "AB." Tables, graphs, and signficance patterns all bear upon the three kinds of effects we have been talking about--the main effect of A, the main effect of B, and the AB interaction effect.

We'll assume that A has two levels, a1 and a2; that B has two levels, b1 and b2.

We'll examine eight different cases. In each case we'll present the table of means, the graph, and significance pattern and tell how they bear on each other. Let's examine our first case in which the main effect of A is signficant, but neither the main effect of B nor the AB interaction is significant.

Notation for Signficance Pattern. A represents the main effect of A. B represents the main effect of B. And AB represents the interaction effect between A and B.

Stars and Dashes. To describe the signficance pattern I will put a little star (*) next to any effect that is signicant and a little dash (--) next to any effect that is not significant.

First Signficance Pattern: Only the Main Effect of A is signficant. Using the above notation, the first illustration shows a significance pattern in which only the main effect of A is significant. The main effect of B is not significant. The AB interactin is not significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

One possible example. The data presented in the table and graphs of the illustration and discussed below represents only one possible way of getting the signficance pattern. It is non-unique; many other possible examples could be made up that would yeild the same significance pattern.

Table of Means. In the upper left of the illustration is a table of means. Just as in previous cases in this lecture the table has cell means, row means, and column means. The levels of A go across the columns; the levels of B go down the rows.

Signficance Pattern. The significance pattern is shown in a little red box just below the table of means. In this case A has a star next to it indicating that the main effect of A is signficant, B and AB have dashes indicating that they are not significant.

Examine the Column means for the Main Effect of A. The main effect of A is found by examing the column means. The column mean for a1 is 20 and the column mean of a2 is 10. We can see that, averaged over all the levels of b, the difference between a1 and a2 is 20 - 10 = 10. Thus it appears that there is a main effect of A, which corresponds to the fact that A is signficant

Examine the Row means for the Main Effect of B. The main effect of B is found by examining the row means. The row mean for b1 is 15 which is exactly the same as the row mean for b2. The main effect of B is 15 - 15 = 0. So the table shows that there is no effect of B, which corresponds to the fact that the main effect of B is not signficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is exactly the same at b1 (top row) as it is at b2 (2nd row). That is, in both the b1 row and the b2 row moving from a1 to a2 drops the score from 20 to 10. So the effect of A does NOT depend on the level of B. So there is no AB interaction. This corresponds to the fact that the AB interaction is not significant.

Graph of the the AB interaction. There are three graphs. The top graph shows the AB interaction. The interaction graph is created graphing the cell means from the table. Along the horizontal axis are the levels of A (a1 and a2). In the graph the blue line is b1 and the red line is b2. Notice that when the cell means are graphed the two lines (b1 and b2) and fall right on top of each other. The blue b1 line goes from 20 to 10 and so does the red b2 line. So clearly the effect of A is the same at b1 and b2, and visually there is no interaction.

In general, if you think about it, parallel lines correspond to a lack of interaction. In this case, not only are they parallel, they fall right on top of each other and are essentially indistinguishable.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see a drop from a1 to a2 of 10 points, indicating a a main effect of A, which is what showed up in the significance pattern.

Graph of the Main Effect of B. If we graph the row means (lower graph on the right) we see that the line is flat because the row mean for b1 is equal to the row mean for b2. (The are both equal to 15.). This flat line indacates that there is no main effect of Be which what we found with the significance test.

Learning Focus. The focus of this part of the lecture is upon your learning to be able to relate various ways of describing and presenting results to each other. A useful goal is to learn to feel it is natural and easy to understand how a graph goes with a table of means and how they both relate to patterns of signficance.

Second Signficance Pattern: Only the Main Effect of B is signficant. The next illustration shows a significance pattern in which only the main effect of B is significant. The main effect of A is not significant. The AB interactin is not significant.

What kinds of tables of means and graphs would go with such a signficance pattern? Once again, the example presented below is only one of many possible data sets that might yeild this signficance pattern.

Examine the Column means for the Main Effect of A. The column mean for a1 is 20 and the column mean of a2 is 10. We can see that, averaged over the two levels of b, the difference between a1 and a2 is 20 - 10 = 10. Thus it appears that there is a main effect of A, which corresponds to the fact that A is signficant

Examine the Row means for the Main Effect of B. The row mean for b1 is 10 and the row mean for b2 is 20. So the main effect of B is 10 - 20 = -10. So the table shows that there is a main effect of B, which corresponds to the fact that the main effect of B is signficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of B is the same at both levels of A. That is, at a1 (first column) the effect of B is 10 - 20 = -10. At a2 (second column) the effect of B is 10 - 20 = -10 also. So the effect of B does NOT depend on the level of A. This corresponds to the fact that the AB interaction was not found to be significant.

We could make a similar and redundant argument from the perpective of A. Notice that the effect of A is zero at both levels of B--b1 (top row) the effect of A is 10 - 10 = 0; at b2 (bottom row) the effect of A is 20 - 20 = 0. That is, in both the b1 row and the b2 row moving from a1 to a2 produces no effect. So the (lack of) effect of A does NOT depend on the level of B. So there is no AB interaction. This corresponds to the fact that the AB interaction was not found to be significant.

Graph of the the AB interaction. The interaction graph is created graphing the cell means from the table. As usual, along the horizontal axis are the levels of A (a1 and a2). In the graph the blue line is b1 and the red line is b2. Notice that when the cell means are graphed the two lines (b1 and b2) are parallel. The blue b1 line goes from 10 to 10 and the red b2 line goes from 20 to 20. So clearly the effect of A is the same at b1 and b2, and visually there is no interaction.

Another way to think about this is to notice that the effect of B is -10 (i.e., 10 - 20 = -10) at a1. The effect of B is also -10 at a2. So the effect of B does not depend on which level of A we are talking about. Thus there is no interaction.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a flat line--the column mean for a1 is 15 and the column mean for a2 is 15. Thus the graph shows no main effect of A, which fits with the lack of a signficant main effect of A.

Graph of the Main Effect of B. If we graph the row means (lower graph on the right) we see that the line angles up from b1 to b2, indicating higher scores at b2 than at b1. So the graph indicates a main effect of B which what we found with the significance test.

Third Signficance Pattern: Both the Main Effects of A and B are signficant. The next illustration shows a significance pattern in which the main effect of A and B are both significant. The AB interactin is not significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

Examine the Column means for the Main Effect of A. In the table of means in the upper left, the main effect of A is found by examining the column means. The column mean for a1 is 15 which is exactly the same as the column mean for a2. The main effect of A is 15 - 15 = 0. So the table shows that there is no effect of A, which corresponds to the fact that the main effect of A is not signficant.

Examine the Row means for the Main Effect of B. The main effect of B is found by examining the row means. The row mean for b1 is 10 and the row mean for b2 is 20. So the main effect of B is 10 - 20 = -10. So the table shows that there is a main effect of B, which corresponds to the fact that the main effect of B is signficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is exactly the same at b1 (top row) as it is at b2 (2nd row). That is, at b1 the effect of A is 15 - 5 = 10. At b2 the effect of A is 25 - 15 = 10. So the effect of A does NOT depend on the level of B. So there is no AB interaction. This corresponds to the fact that the AB interaction was not found to be significant.

Graph of the the AB interaction. As usual, the interaction graph (top graph in the illustration) is created graphing the cell means from the table, with the horizontal axis indicating the levels of A (a1 and a2) and lines in the graph (red and blue) indicating levels of B. Notice that when the cell means are graphed the two lines (b1 and b2) are parallel. The blue b1 line drops 10 points, going from 15 to 5. And the red b2 line also drops 10 points, going from 25 to 15. So the graph shows visually that there is no interaction, the effect of A is to produce a 10 point drop at both levels of B.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a line that drops from 20 to 10, indicating a main effect of A, which fits with the signficant main effect of A.

Fourth Signficance Pattern: All effects signficant. The next illustration shows a significance pattern in which the main effect of A and B are both significant and the AB interactin is also significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

Examine the Column means for the Main Effect of A. The column mean for a1 is 20 and the column mean for a2 10. The main effect of A is therefore 20 - 10 = 10. So the table shows that there is a main effect of A, which corresponds to the fact that the main effect of A is signficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is DIFFERENT AT DIFFERENT LEVELS of B. At b1 (top row) A has no effect (10 - 10 = 0). In contrast, at b2, the effect of A is 20 (i.e., 30 - 10 = 20). The effect of A very much depends on level of B--A has no effect at b1 and a strong effect at b2. So there IS an AB interaction. This corresponds to the fact that the AB interaction was found to be significant.

Graph of the the AB interaction. As usual, the interaction graph (top graph in the illustration) is created graphing the cell means from the table. The horizontal axis indicates the levels of A (a1 and a2) and the lines in the graph (red and blue) indicate levels of B. Notice that when the cell means are graphed the two lines (b1 and b2) are NOT parallel; rather, they start far apart at a1 and touch at a2. The blue b1 line is level; it goes from 10 to 10. In contrast, the red b2 line drops 20 points, going from 30 to 10. So the graph shows visually that there is an interaction, the effect of A is to produce a 20 point drop at one level of B and no drop at all at the other level of B.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a line that drops from 20 to 10, indicating a main effect of A, which fits with the signficant main effect of A.

Integrating signficance patterns with data patterns. In this fourth case the significance pattern resulting from a 2-Way ANOVA produced a significant F for the main effect of A, significant F for the main effect of B, and a significant F for the AB interaction. We've examined how this signficance pattern is reflected in the table of means and varirous ways of graphing those means. Now typically in a journal article, you don't get all three of these presentations, you'll just get one, perhaps two. So it is important to practice how these expressions of results are related to each other so that you can think critically about what the results mean.

Fifth Signficance Pattern: A and AB Signficant; B not signficant. The next illustration shows a significance pattern in which the main effect of A and the AB interaction are both significant but the B main effect is not significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

Examine the Row means for the Main Effect of B. The row mean for b1 is 15 and the row mean for b2 is 15. So the main effect of B is 15 - 15 = 0. So the table shows that there is no main effect of B, which corresponds to the fact that the main effect of B is not signficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is DIFFERENT AT DIFFERENT LEVELS of B. At b1 (top row) A has no effect (15 - 15 = 0). In contrast at b2 (second row) the effect of A is 20 (i.e., 25 - 5 = 20). The effect of A very much depends on level of B--A has no effect at b1 and a strong effect at b2. So there IS an AB interaction. This corresponds to the fact that the AB interaction was found to be significant.

Graph of the the AB interaction. As usual, the interaction graph (top graph in the illustration) is created graphing the cell means from the table. Notice that when the cell means are graphed the two lines (b1 and b2) are NOT paralle. The blue b1 line is level; it goes from 10 to 10. In contrast, the red b2 line drops 20 points, going from 25 to 5. So the graph shows visually that there is an interaction, the effect of A is to produce a 20 point drop at one level of B and no drop at all at the other level of B.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a line that drops from 20 to 10, indicating a main effect of A, which fits with the signficant main effect of A.

Graph of the Main Effect of B. If we graph the row means (lower graph on the right) we see that the line is level going from b1 to b2. So the graph indicates a no main effect of B which what we found with the significance test.

Integrating signficance patterns with data patterns. In this fifth case the significance pattern resulting from a 2-Way ANOVA produced a significant F for the main effect of A, a nonsignificant F for the main effect of B, and a significant F for the AB interaction. We've examined how this signficance pattern is reflected in the table of means and varirous ways of graphing those means. Now typically in a journal article, you don't get all three of these presentations, you'll just get one, perhaps two. So it is important to practice how these expressions of results are related to each other so that you can think critically about what the results mean.

Sixth Signficance Pattern: B and AB Signficant; A not signficant. The next illustration shows a significance pattern in which the main effect of B and the AB interaction are both significant but the A main effect is not significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

Examine the Column means for the Main Effect of A. The column mean for a1 is 15 and the column mean for a2 15. The main effect of A is therefore 15 - 15 = 0. So the table shows that there is a no main effect of A, which corresponds to the fact that the main effect of A is not signficant in the 2 Way ANOVA.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is DIFFERENT AT DIFFERENT LEVELS of B. At b1 (top row) A has an effect = +10 (15 - 5 = 10). In contrast at b2 (second row) the effect of A is MINUS 10 (i.e., 15 - 25 = -10). The effect of A very much depends on level of B--A has OPPOSITE effects at b1 and b2. So there IS an AB interaction. This corresponds to the fact that the AB interaction was found to be significant.

Graph of the the AB interaction. Notice that when the cell means are graphed the two lines (b1 and b2) are NOT parallel--they move in opposite directions. As it goes from a1 to a2, the blue b1 line drops 10 points from 15 to 5. In contrast, the red b2 line rises 10 points, going from 15 to 25. So the graph shows visually that there is an interaction, the effect of A is to produce a 10 point rise at one level of B and a 10 point drop at the other level of B.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a line that is level, which fits with the nonsignficant main effect of A.

Graph of the Main Effect of B. If we graph the row means (lower graph on the right) we see that the line rises going from b1 to b2. So the graph indicates a main effect of B which what we found with the significance test.

Integrating signficance patterns with data patterns. In this sixth case the significance pattern resulting from a 2-Way ANOVA produced a nonsignificant F for the main effect of A, a significant F for the main effect of B, and a significant F for the AB interaction. We've examined how this signficance pattern is reflected in the table of means and varirous ways of graphing those means.

Seventh Signficance Pattern: AB Signficant; A not signficant and B not signficant. The next illustration shows a significance pattern in which the main effects of A and B are both not signficant while the AB interaction is significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

Examine the Row means for the Main Effect of B. The row mean for b1 is 15 and the row mean for b2 is 15. So the main effect of B is 15 - 15 = 0. So the table shows that there is not a main effect of B, which corresponds to the fact that the main effect of B is nonsignficant.

Examine the Cell means for the AB Intereaction Effect. Notice that the effect of A is DIFFERENT AT DIFFERENT LEVELS of B. At b1 (top row) A has an effect = +10 (20 - 10 = 10). In contrast at b2 (second row) the effect of A is MINUS 10 (i.e., 10 - 20 = -10). The effect of A very much depends on level of B--A has OPPOSITE effects at b1 and b2. So there IS an AB interaction. This corresponds to the fact that the AB interaction was found to be significant.

Graph of the the AB interaction. Notice that when the cell means are graphed the two lines (b1 and b2) are NOT parallel--they move in opposite directions. As it goes from a1 to a2, the blue b1 line drops 10 points from 20 to 10. In contrast, the red b2 line rises 10 points, going from 10 to 20. So the graph shows visually that there is an interaction, the effect of A is to produce a 10 point rise at one level of B and a 10 point drop at the other level of B.

Graph of the Main Effect of A. The main effect of A shows up in a graph of the columrn means (lower graph on the left in the illustration). You can see that moving from a1 to a2 produces a line that is level, which fits with the nonsignficant main effect of A.

Graph of the Main Effect of B. If we graph the row means (lower graph on the right) we see that, as with A, the line is level going from b1 to b2. So the graph indicates no main effect of B which what we found with the significance test.

Integrating signficance patterns with data patterns. In this seventh case the significance pattern resulting from a 2-Way ANOVA produced a nonsignificant F for the main effect of A, a nonsignificant F for the main effect of B, and a significant F for the AB interaction. We've examined how this signficance pattern is reflected in the table of means and varirous ways of graphing those means.

Eighth and Final Signficance Pattern: Nothing is signficant. The last illustration shows a significance pattern in which the main effects of A and B are both not signficant and the AB interaction is also not significant.

What kinds of tables of means and graphs would go with such a signficance pattern?

A, B, and AB have no effect. If you examine the table and graphs in the illustration you will see that they all show there no effects for A, B and AB.