wEB aPPENdIX fOR rESEARCH DESIGN EXPLAINED Interpreting ordinal and DisORDINAL INTERACTIONS

© 2006-2016, Mark Mitchell and Janina Jolley.

Interpreting Ordinal and Disordinal interactions

You do a 2 X 2 experiment and get an interaction. How do you make sense of that interaction?

The easiest way to make sense of it is to graph your two simple main effects.[1] Because you have a significant interaction, the lines representing these simple main effects will have different slopes, reflecting the fact that the treatment appears to have one effect in one condition, but a different effect in another condition. In other words, because you have a significant interaction, the lines in your graph will not be parallel.

Closer inspection of these nonparallel lines will tell you what type of interaction you have. As you’ll see, it is important to know whether you have an ordinal interaction or a disordinal (crossover) interaction.

Ordinal Interactions May Mean That the Amount of Effect One Treatment Has Depends on the Level of the Other Treatment

Suppose that your nonparallel lines are sloping in the same direction but do not actually cross. For example, both of the lines may slope upward, but one of the lines has a steeper slope. Or, both of the lines may slope downward, but one has a steeper slope. In either case, you have an ordinal interaction.

Ordinal interactions suggest that a treatment has a more intense effect in one condition than another. For example, you would find an ordinal interaction if a treatment was more effective for one type of patient than for another. You would also find an ordinal interaction if a teaching strategy helped both shy and outgoing children but helped outgoing children more. In the Figure below, you can see another example of an ordinal interaction: Exercise boosted participants’ calorie consumption by only 200 calories in the no-caffeine condition, but it boosted calorie consumption by 800 calories in the caffeine condition.

In this case, the ordinal interaction seems to suggest that combining two independent variables  (caffeine and exercise) produces larger than expected effects. The combination is greater than the sum of its parts.

In other cases, ordinal interactions suggest that combining treatments is less effective than you would expect from knowing the factors’ individual effects. That is, some ordinal interactions result from the combination of treatments being less than the sum of the effects of the individual treatments.

To sum up, some ordinal interactions reflect the fact that a treatment
has more of an effect when combined with another treatment. Some ordinal interactions reflect the fact that a treatment has more of an effect when another treatment is not around. All ordinal interactions suggest that a factor appears to have more of an effect in one condition than in another condition.

Ordinal Interactions May Be Measurement-Induced Mirages

We say appears to have more of an effect because it is not easy to determine whether a variable had more of a psychological effect in one condition than in another. For example, to state that the difference between 2,200 calories and 2,000 calories is less than the difference between 3,000 calories and 2,200 calories, you must have at least interval scale data: data in which the differences between consecutive numbers always represent the same psychological difference. For example,  the difference between a score of “1” and a score of “2” on the measure must be the same—in terms of the characteristic that you are trying to measure— as the difference between 6 and 7. (To learn more about interval scale data, see Chapter 6.)

If you are interested only in number of calories consumed, you have interval scale data. However, if you are using calories consumed as a measure of how hungry people felt, your measure may not be interval. Specifically, if you do not have a one-to-one correspondence between number of calories consumed and degree of perceived hunger, your data are not interval. Instead, your data are probably ordinal: higher scores indicate more of a quality but equal differences between scores do not necessarily indicate equal differences in the characteristic allegedly being measured. For example, it may take the same increase in perceived hunger to make a person who normally eats 2,000 calories consume an additional 200 calories as it does to get someone who would normally eat 2,200 calories to eat an additional 800 calories. If it does, then the ordinal interaction depicted in the previous graph is an artifact (an unintended result) of calories consumed being an ordinal, rather than an interval, measure of hunger. In that case, if you had used an interval scale measure of hunger, you would not have obtained an interaction.

To visualize how scale of measurement can affect our findings, consider the table below. The first couple of rows of that table are quite similar to the results we just discussed. For example, if you look at calories consumed, you see that there is an ordinal interaction: The simple main effect of exercise in the no caffeine condition (200) is less than the simple main effect of exercise in the caffeine condition (800). This interaction may be due to exercising boosting feelings of hunger more in the caffeine condition than in the non-caffeine condition. If this is the case, if we repeated the study using a 1-9, interval rating scale measure of hunger, we would again find an interaction. We have depicted that state of events in the third row of the table. However, what if calorie consumption doesn’t map so nicely onto hunger? In that case, as we shown in the last row of the table, when we used an interval scale measure of hunger, we might not obtain an interaction. Instead, we might  find that the main effect of exercise would be the same in both the no caffeine and caffeine conditions.

 No caffeine/ No exercise No caffeine/ Exercise Caffeine/ No exercise Caffeine/ Exercise 1st simple main effect 2nd simple main effect Calories consumed 2,000 2,200 2,400 2,600 2,800 3,000 2,200 – 2,000 = 200 3,000 – 2,400 = 600 Possible hunger scores on a 1-9 scale if calories are an interval measure of hunger 4 5 6 7 8 9 5 – 4 = 1 9 – 6  = 3 Possible hunger scores on a 1-9 scale if calories are not an interval measure of hunger 4 5 5.33 5.66 5.99 6.33 5 - 4 = 1 6.33 – 5.33 = 1

To see how you might get an ordinal interaction even when the effect of combining two variables produces nothing more than the sum of their individual effects, look at figures “A” and “B” below. In Figure (A),  the lines are not parallel and thus indicate an interaction. Now, look at the bottom Figure (B), which is a graph of the same data. In Figure B, the lines are parallel, indicating no interaction.

Why does one graph suggest an interaction whereas the other does not? The first graph depicts an interaction because it, like most graphs you have seen, makes the distance between 3 and 4 equal to the distance between 7 and 8. However, by doing this, the author of the graph assumes that these data are interval. Thus, the difference in hunger between those participants who rate their hunger a “3” and those participants who rate their hunger a “4” is depicted as being the same as the difference in hunger between those who rate their hunger a “7” and those who rate their hunger an “8.” If this assumption is true, then caffeine makes participants feel hungrier in the exercise condition than it does in the no-exercise condition.

The second graph (B) shows what can happen if we, rather than buying the assumption that these data are interval, suppose that we have ordinal data. Specifically, the graph shows what happens when the difference between a “7” and an “8” rating is greater—in terms of how much hunger people feel—than the difference between a “3” and a “4” rating.  In that case, at the psychological level, caffeine’s effect on the subjective state of “feeling hungry” is the same in the exercise condition as it is in the no-exercise condition. Thus, even though there is an interaction at the statistical level, there is not an interaction at the psychological level: Caffeine has the same psychological effect in both conditions. In other words, seeing that the treatment makes a greater change in participants’ scores in one condition doesn’t necessarily mean that the treatment makes a greater change in participants’ feelings.

When to Suspect That an Ordinal Interaction Is an Artifact of Having Ordinal Data

As you have seen, ordinal interactions may be nothing more than an illusion caused by having ordinal, rather than interval, data. Because you can rarely be sure that you have interval data, you should always be cautious when interpreting ordinal interactions. However, you should be especially cautious if you have reason to believe that your data are ordinal.

When Data are Ranks or Scores on Some Other Ordinal Measure

If your data come from having participants rank items from lowest to highest, you have ordinal data. You can’t say that the difference between something ranked “1” and something ranked “2” is the same as the difference between something ranked “3” and something ranked “4” (see Chapter 6).

When  Scores Suggest that Either Ceiling or Floor Effects Are Likely

Because ordinal data make ordinal interactions difficult to interpret, you may want to avoid using ordinal measures. Unfortunately, however, avoiding ordinal data is not as easy as avoiding ordinal measures. Even a measure that seems like it should provide interval data may end up providing ordinal data.

To understand how an interval scale measure could produce ordinal data, imagine a typical bathroom scale. Normally, it would provide interval data. However, what if you were measuring football players on it? Your scale would provide interval measurement to those players who weighed less than 250 pounds (or whatever your scale went up to). However, everyone 250 and above would, according to your scale, weigh 250. Thus, you would know that someone who, according to your scale, weighed 250 was heavier than someone weighing 245, but you wouldn’t know how much heavier.

Not only would your scale fail you at extremely high weights, it would also fail you at extremely low weights. Thus, if you were weighing the food on football players’ plates, your scale would not provide accurate weights.

In technical terminology, your bathroom scale has two weaknesses. First, its ceiling (the highest score participants can receive) of 250 pounds is too low to weigh some of the heavier football players. Second, its floor (the lowest score participants can receive) of one pound is too high to accurately measure objects less than 1 pound.

The scale’s low ceiling could hide the effect of a treatment. For example, suppose the heavy football players are put on a weightlifting program that makes them even heavier. Although the treatment works, the results would not show up on the bathroom scale: The heavy players would, according to that scale, still “weigh” 250 pounds. In technical jargon, your results are misleading because of a ceiling effect: the effect of a treatment or combination of treatments is underestimated because the dependent measure is not sensitive to values above a certain level.

The scale’s high floor could also hide the effect of a treatment. For example, suppose the football players are rewarded for putting less food on their plates, and the reward system works. However, the researcher measures the amount of food on the plate by using a bathroom scale that can’t accurately weigh anything under 5 pounds. The measure’s high floor makes it look like the players aren’t eating less. In technical jargon, our results are misleading because of a floor effect: The effects of the treatment or combination of treatments is underestimated because the dependent measure places too high a floor on what the lowest response can be.

Ceiling and floor effects can make it seem like an effective treatment has no effect. In addition, ceiling and floor effects can make it seem like an effective treatment has a strong effect for one group but no effect for another group.

Let’s see how a ceiling effect could make it look like a treatment that is equally effective for two groups is only effective for one of the groups. Let’s start by supposing that both light and heavy football players are put on a weightlifting program. Furthermore, suppose that both groups gain 20 pounds. The light football players go from 180 to 200 pounds; the heavy players go from 300 to 320. According to the bathroom scale, the light players gain 20 pounds (180–200), but the heavy football players haven’t gained a pound. The scale still has them all weighing 250 pounds. In this case, a ceiling effect made it look like weightlifting has more of an effect on light than on heavy players, even though weightlifting’s effect is the same for both types of players. That is, due to a ceiling effect, our scale gave us an ordinal interaction when we should not have had one.

Like bathroom scales, psychological scales may be plagued by low ceilings and high floors. Thus, as with extreme scores on our bathroom scale, extreme scores on a psychological measure may be misleading. Some of the low scorers may deserve much lower scores, but the measure is unable to give it to them. In a sense, the “floor” (the lowest score they can receive) is too high. Or, some of the high scorers might, if given a chance, score much higher than the others—but the measure doesn’t give them that chance. In such a case, the “ceiling” (the highest score they can receive) is too low.

Thus far, you know that ceiling and floor effects can contaminate your results. If your measure’s ceiling is too low, ceiling effects may be contaminating your results. If your measure’s floor is too high, floor effects may be contaminating your results. In the next sections, you will learn (a) when to suspect that a ceiling or floor effect is affecting your results and (b) how ceiling and floor effects contaminate your results. Thus, after reading the next section, you will be appropriately cautious when interpreting ordinal interactions.

When to Suspect That an Ordinal Interaction Is Due to Ceiling Effects.

If a group’s average scores are quite high, you should suspect that your ordinal interaction may be due to ceiling effects. Ceiling effects occur when the measure does not allow participants to score as high as they should  For example, in the picture below, the gym’s ceiling is only 10 feet high, so coaching can’t make Larry jump any higher than 10 feet.

To take a more realistic example, imagine an extremely easy knowledge test, in which half the class scores 100%. The problem with such a test is that we can’t differentiate between the students who knew the material fairly well and the students who knew the material extremely well. The test’s “low ceiling” did not allow very knowledgeable students to show that they knew more than the somewhat knowledgeable students.

To see how a ceiling effect can create an ordinal interaction, consider the following experiment. An investigator wants to know how information about a specific person affects the impressions people form of that person. The investigator uses a 2 (information about a stimulus person’s traits [no information versus extremely positive information]) X 2 (information about a stimulus person’s behavior [no information versus extremely positive information]) factorial experiment. For the dependent measure, participants rate the stimulus person’s character on a three-point scale (1 = below average, 2 = average, 3 = above average).

As you can see from the top of the next figure, the investigator obtains an ordinal interaction. The interaction suggests that getting information about a specific person’s behavior has less of an impact if participants already have information about that person’s traits. In fact, the interaction suggests that if participants already know about the stimulus person’s traits, information about the person’s behavior is worthless.

The problem in interpreting this interaction is that the results could be due to a ceiling effect. That is, even if getting additional favorable information about the stimulus person raises participants’ opinions of that person, participants cannot show this increased respect. That is, participants may feel that a person with a favorable trait is a 3 (above average) and a person with both a favorable trait and a favorable behavior is a 4 (well above average)—but they cannot rate the person a 4. The highest rating they can give a person is a 3. The highest rating (above average) on this scale—the ceiling response—is not high enough.

By not allowing participants to rate the stimulus person as high as they wanted to, the investigator did not allow participants to rate the positive trait/positive behavior person higher than either the positive trait/no behavior person or the positive behavior/no trait person. Thus, the investigator’s ordinal interaction was due to a ceiling effect. In other words, the interaction is due to the dependent measure placing an artificially low ceiling on how high a response can be. As you can see from the bottom part of that graph, “raising the ceiling” would eliminate some ordinal interactions.

When to Suspect that an Ordinal Interaction Is Due to Floor Effects.

Just as ceiling effects can account for ordinal interactions, so can their opposites—floor effects. For example, suppose the investigator uses the same three-point rating scale as before (1 = below average, 2 = average, 3 = above average). However, instead of using no information and extremely positive information, the investigator uses no information and extremely negative information. The investigator might again obtain an ordinal interaction (see the next figure). Again, the interaction would indicate that adding behavioral information to trait information has little effect on participants’ impressions. This time, however, the interaction could be due to the fact that participants could not rate the stimulus person lower than a “1” (below average). The problem is that the bottom rating—the “floor”—is too high.

By not allowing participants to rate the person as low as they wanted to, the investigator did not allow participants to rate the negative trait/negative behavior stimulus person lower than the negative trait/no behavior person or the negative behavior/no trait person. Thus, the investigator’s ordinal interaction was due to a floor effect. As you can see from the bottom half of the previous, “lowering the floor” can eliminate some ordinal interactions.

As floor and ceiling effects show, an ordinal interaction may reflect a measurement problem rather than a true interaction (see next figure). So, be careful when interpreting ordinal interactions.

Disordinal (Crossover) Interactions Mean That The Type of Effect One Treatment Has Depends on the Level of the Other Treatment

You do not have to be so careful if you have a crossover interaction. When you have a crossover interaction, as the term “crossover interaction” suggests, the lines in your graph actually cross.

Crossover interactions often indicate that a factor has one kind of effect in one condition and the opposite kind of effect in another condition. For example, you would have a crossover interaction if a therapy that helps patients who have one kind of problem actually hurts patients who have a different kind of problem. In the next figure, you can see another example of a cross-over interaction: In the no-caffeine condition, exercise increases calorie consumption, but, in the caffeine condition, exercise decreases calorie consumption.

Crossover interactions are also called disordinal interactions because, unlike ordinal interactions, they can’t be an artifact of having ordinal, rather than interval, data. To understand why disordinal interactions can’t be due to having ordinal data, let’s imagine a factorial experiment designed to determine the effect of watching exercise videos and monitoring what one eats on physical fitness. Specifically, middle-aged adult volunteers are assigned to one of four conditions: (1) no video/no diet monitoring, (2) no video/diet monitoring, (3) video/no diet monitoring, and (4) video/diet monitoring. After 6 weeks, fitness is assessed.

The experimenter assesses fitness by having the adults start one block east of First Street and seeing how many blocks the person can run in 10 minutes. Thus, if a person makes it to First Street, he has run one block, if he makes it to Third Street, he gets credit for running three blocks.

Unfortunately, the experimenter did not check to make sure that all the blocks are equal in length. Indeed, as it turns out, the blocks vary considerably in length.

Because some blocks are very short and others are fairly long, the measure provides only ordinal data. For example, we would know that someone who ran eight blocks ran farther than someone who ran six blocks, but we wouldn’t know how much farther. We would know one ran two blocks farther, but two blocks might be 100 feet or it might be 10,000 feet. Therefore, if Mary runs eight blocks to Mabel’s six, and Sam runs four blocks to Steve’s two, we cannot say that Mary outran Mabel by the same distance as Sam outran Steve. Mary may have run 100 feet more than Mabel, but Sam may have run 10,000 feet more than Steve.

How will the use of an ordinal measure hurt the experimenter’s ability to interpret the results? The answer to this question depends on the pattern of the results.

If the researcher gets the ordinal interaction depicted in the next figure, there is a problem because the apparent interaction may really be just an artifact of having ordinal data. For example, the interaction could be due to the distance between Seventh and Eighth Streets being longer than the distance between Third and Fifth Streets. Thus, if the researcher had used an interval measure (number of feet run), the researcher might not have found an interaction.

If, on the other hand, the researcher gets the disordinal interaction depicted in the next  figure, this interaction can’t be due to having ordinal data. Even if the blocks are all of different lengths (not at all interval), the interaction would still demonstrate that weight-monitoring was more effective for the people watching the exercise videos than for those not watching the videos.

Regardless of the length of the blocks, we know that the distance between Third Street and Eighth Street is greater than the distance between Fifth Street and Seventh Street. (We know this because the distance between Third and Eighth includes the distance between Fifth and Seventh. As you can see from the table below, to walk from Third to Eighth, you have to walk through Fifth and Seventh.)

 Street 1st 2nd 3rd 4th 5th 6th 7th 8th

Conclusions About Interpreting Ordinal and Disordinal Interactions

To reiterate, disordinal interactions are less difficult to interpret than ordinal interactions. Ordinal interactions are difficult to interpret because there are always two possible explanations for an ordinal interaction.

One possible explanation is that the ordinal interaction really represents the fact that combining your two independent variables produces an effect that is different from the sum of their individual effects. Specifically, combining the factors really has either less of, or more of, an effect than the sum of their individual effects.

The second possibility is that the apparent interaction is merely an artifact of ordinal scale measurement (see Chapter 6 for a review of scales of measurement). If you had more accurately measured the construct, you would not have found an interaction. Therefore, if you want to say that an ordinal interaction represents a true interaction, you need to establish that you have interval or ratio data.

To illustrate the difficulty of interpreting an ordinal interaction, suppose you find that a treatment boosted scores from 15 to 19 in one condition, but from 5 to 8 in the other. At the level of scores, you have an interaction: The treatment increased scores more in one condition than in the other. But what about at the level of psychological reality? Can you say that going from a 15 to a 19 represents more psychological change than going from a 5 to an 8? Only if you have interval or ratio scale data. In other words, with an ordinal interaction, you can only conclude that the variables really interact if you can say that a one-point change in scores at one end of your scale is the same thing (psychologically) as a one-point change at any other part of your scale.

With crossover (disordinal) interactions, on the other hand, you are not comparing differences between scores on one part of your scale with differences between scores on an entirely different part of your scale. Instead, you are making comparisons between scores that overlap. For example, with a crossover interaction, you might only have to conclude that the psychological difference between 10 and 30 is bigger than a difference between 15 and 19. Because the difference between 10 and 30 includes 15–19, you can conclude that the difference between 10 and 30 is bigger—even if you only have ordinal data. Thus, when you have a crossover (disordinal) interaction, you can conclude that your variables really do interact.

[1] The simple main effect of an independent variable refers to the effects of that  factor at a specific level of a second independent variable. The simple main effect could have been obtained merely by doing a simple experiment. It can be estimated by comparing the mean for one group with the mean for a second group. For instance, the simple main effect for caffeine in the no-exercise conditions could be estimated by comparing the no-caffeine, no-exercise group mean to the caffeine, no-exercise group mean.