TheOneWayANOVA
So you took Stats I and Stats II passed. But do you remember what you did, how you did it, and why you did it? If you need some basic statistic reminders for the One Way ANOVA, then this is the lecture for you! I am going to talk about a One-Way ANOVA example in this document that corresponds to the same example you saw in the Descriptive Statistics Crash Course (#1) and t-Test Crash Course (#2) where participants were asked to recall how much money they spent on textbooks the prior semester. However, for this ANOVA crash course, we are going to add a third condition: control (a third group of participants who do not see any prior book recall amounts on the list). As you can see here, we have ONE independent variable (hence the One Way ANOVA), but here we have three levels (or three conditions): High, Low, and Control. The good news is that this mini-lecture will sum up the basics of the ANOVA for you as we look at this study, but you can find additional information about the ANOVA in your textbooks. On the final pages of this document are several questions based on this crash course. Answer these questions, and then go into your “Crash Course in Statistics – The One-Way ANOVA Quiz #3” in your Canvas assessments menu and copy over your answer. Each Crash Course Quiz counts 5 points.
How, when, and why do a One Way ANOVA?
Before we get to the example, let me give you some basic information about the One Way ANOVA. Do you recall the t-Test, where we compared two means to see whether and in what direction the means differed? Well, a One Way ANOVA is very similar, but here we compare three or more means to see if they differ significantly from one another. In this analysis, we need three pieces of information: 1) the means for each of the three groups (descriptive statistics), 2) the One-Way ANOVA information itself, and 3) post hoc tests.
1). Once again, remember that a mean is the average score for that condition. That is, you add up all of the scores in a condition and divide by the number of total scores to arrive at the average. Since a One Way ANOVA looks at three or more different conditions, we have at least three means: one for each condition. The means here (plus the standard deviation, which we will talk about in the lecture) are descriptive statistics. That is, they help describe the data.
2). The One-Way ANOVA information itself is a test of inferential statistics. That is, we infer significant differences between the three or more groups. When writing it out, you will see a very common layout for the One Way ANOVA, something like: F(2, 134) = 2.61, p = .021. The F tells you this is a One Way ANOVA. The 2 and 134 tells us our degrees of freedom (more on that in out lecture). The 2.61 is the actual number for the One Way ANOVA. The p indicates whether it is significant (if it is less than .05, then it is significant).
3). Finally, we have to consider post hoc tests. You might recall using the Tukey post hoc test in the past, but do you remember why you used it? Take a step back and think about the t-Test, which looked at two means: Mean A and Mean B. If Mean A is 4.56 and Mean B is 7.67 and your t-Test is significant (that is, p is less than .05), then you simply compare the two means to see which is higher: Mean A or Mean B. Here, Mean B is clearly higher (7.67 is higher than 4.56), and since the t-Test is significant then Mean B is significantly higher than Mean A. But when we have three levels to our independent variable, we are now dealing with three means: Mean A, Mean B, and Mean C. Let’s say Mean A is 4.56, Mean B is 7.67, and Mean C is 6.21. If our One Way ANOVA is significant (that is, it is less than .05), we know the means differ. The question is, which of the three means differ? Does Mean A differ from Mean B? Does Mean B differ from Mean C? Does Mean A differ from Mean C? Or there might be other combinations. Maybe Mean A and Mean C do not differ from each other, but both are significantly lower than Mean B. Unlike the t-Test, we don’t know which of the three means differ, which is why we run a post hoc test (like Tukey) to compare Mean A to Mean B, and Mean A to Mean C, and Mean B to Mean C. It runs all of those analyses for us in one test. So you might wonder, “Why not just run three t-Tests, with one t-Test comparing Mean A to Mean B, a second t-Test comparing Mean B to Mean C, and a third t-Test comparing Mean B to Mean C.?” Well, you could actually do that, but we run into a Type I error. That is, the more tests we run, the greater the chance one of them will be significant. If we run three t-Tests, we open up the chance of one of them being falsely positive. With the One Way ANOVA, we just run the one test to compare the three means (note that the post hoc tests are still a part of the One Way ANOVA – it compares the three means under the umbrella of the One-Way ANOVA test).
Like the t-Test, we run a One Way ANOVA only under certain conditions.
First, our dependent variable (the variable we measure) must be continuous / scaled. That is, the DV has to be along a scale. For example, it can be an attitude (“On a scale of 1 to 9, how angry are you?”), a time frame (“How quickly did the salesperson help the customer on a scale of zero seconds to a thousand seconds?”), or money estimation (“How much do you recall spending on textbooks last semester?”). We call these interval or ratio scales, which means we can use a t-Test or ANOVA. We CANNOT run a One Way ANOVA on categorical data. That is, if we have a yes / no question (“Are you lonely: Yes or No”) or a category-based question (“What is your favorite food: hamburgers, pizza, salad, or tacos?”), then we cannot run a One Way ANOVA. These latter questions are based more on choice of option rather than an actual rating scale, and thus we cannot use a One Way ANOVA on them.
Second, we run a One Way ANOVA when we have only one independent variable and that independent variable has at least three conditions (Note: it can have more than three levels, but you still only have one independent variable). That is, we compare the means from Condition A, Condition B, and Condition C. If our One Way ANOVA is significant (p is less than .05), then we look at our post hoc tests to see which means differ. “
The One Way ANOVA
was significant, F(2, 134) = 2.61, p = .021. Tukey post hoc tests showed that Condition A (mean = 48.38) was significantly lower than Condition B (mean = 63.25). In addition, Condition C (mean = 48.88) was significantly lower than Condition B. However, Condition A did not differ significantly from Condition C.”
Let’s see how this looks using the textbook money example.
Textbook Study – How Much Did You Spend On Textbooks (High, Low, or Control)
Recall the basic set-up for our money spent on textbooks. Researchers ask participants to recall how much they spent on textbooks the prior semester, and has each participant write their answer on a survey sheet. In two conditions, the first ten answer slots are already filled in, presumably by other respondents. However, the researcher actually completed those ten slots, and manipulated the dollar amounts so that in in the High Dollar Condition, the dollar amounts ranged from $
350
to $450 (Figure 1). In the Low Dollar Condition, amounts ranged from $
250
to $350 (Figure 2). In our new study, the researcher provides a third condition in which there are no prior dollar amounts on the list (Control Condition – Figure 3). Using psychological principles based on conformity and informational social influence (e.g. participants relying on the behavior of other individuals when they lack a clear memory), the researcher predicts that those in the High Dollar Condition will recall spending more money on textbooks the prior semester than those in the Low Dollar Condition, with those in the Control condition providing a dollar amount somewhere in the middle.
Here, the independent variable is Dollar Condition (High versus Low versus Control) while the dependent variable is the amount of money participants recall spending on textbooks (in $). Imagine we have eight real participants in the High Dollar Condition, eight real participants in the Low Dollar Condition, and eight real participants in the Control Condition (and no, we are not including the original researcher-completed dollar amounts on the sheet passed out by the researcher in the High and Low Dollar Conditions, as those are not real participants!).
Figure 3: No Prior Dollar Amount Condition
Consider the data:
|
Condition B (Low) |
Condition C (Control) |
|
275 |
| 350 |
| 325 |
| 275 |
| 250 |
|
260 |
| 300 |
|
315 |
| ∑B = 2350 |
Mean = $337.50 |
Mean = $293.75 |
∑, or the symbol for Sigma, means “the sum of”. Thus ∑A is the sum of the scores for Condition A. That is,
+
+
+
+
+
+
+
= 2700. There are eight scores here, so we divide 2700 / 8 = 337.50, giving us our mean of $337.50 for Condition A (High Dollar Condition). We do the same thing for Condition B (Low Dollar Condition), giving us a mean of $293.75 (2350 / 8 = 293.75). Finally, we do the same thing for Condition C (Control), giving us a mean of $306.25 (2450 / 8 = 306.25).
For the first part of our analysis, we compare the means. As you see, $293.75 in Condition B (Low) looks lower than $337.50 in Condition A (High). Now consider the $306.25 in Condition C (Control), which falls between the High and Low Dollar conditions. That $306.25 doesn’t look that different from the $337.50 in the High Dollar Condition or the $293.75 in the Low Dollar Condition.If I were “eyeballing” this, I would think that participants recall spending significantly more in the High Dollar Condition than in the Low Dollar Condition, but that the Control condition doesn’t differ from either the High or Low Dollar Conditions.However, just because some of our means seem to differ doesn’t mean they do differ. To make that assessment, we run the One Way ANOVA and look at the p value to see if it is p is less than .05. We can do this by hand (like you did in Stats I and Methods One) or we can take the easy road and let SPSS calculate it for us. I am going to take the easy road, but keep in mind that we still have to interpret what SPSS tells us.
For the next section, I am going to open SPSS and run a One Way ANOVA. I’ll use screenshots from SPSS as I go, but feel free to run these analyses yourself. Just set up your SPSS file like mine (I also included this SPSS file for you in Canvas if you prefer to use that. It is called “Crash Course Quiz #3 – Textbook Money (ANOVA Practice)”, but it is a short data set, so I recommend setting up your own SPSS file using the values from the table above). I am just going to give you the basics here, but you can refer to other sources to figure out some of the info we get from the One Way ANOVA not covered in this lecture (like homogeneity of variance, Welch test, etc.).
SPSS – Our Textbook Money Recall Study
Click the button.
Tick the “Descriptive”, and “Means Plot” checkboxes in the Statistics area as shown below:
You will be presented with the following:
- Put the “Condition (1 = High, 2 = Low, 3 = Control)” variable into the “Factor” box and the “How much participants recall spending” variable into the “Dependent List” box by highlighting the relevant variables and pressing the buttons. Note that SPSS uses different names for variables. It calls the dependent variable the “Dependent List” and it calls the independent variable the “Factor”. Just remember that our dependent variable (Factor) must be scaled in order to run this test (1 to 9, or 1 to 5, or even 0 to 100,000). The independent variable must be categorical (dressy v. sloppy v. casual, old v. middle aged v. young, republican v. democrat v. independent, high v. medium v. low, etc.).
Click the button.
Click the button. Then click the button.
Output of the One Way ANOVA in SPSS
You will be presented with several tables containing all the data generated by the One-Way ANOVA procedure in SPSS. Some are useful while others are … less useful (at least to us. They may be more important if you ever submit an article to a journal, but we will ignore them)!
Descriptive Statistics Table (Useful!)
The descriptives table (see below) provides some very useful descriptive statistics including the mean, standard deviation and 95% confidence intervals for the dependent variable (How much participants recalled spending on textbooks) for each separate group (High Dollar Condition, Low Dollar Condition & Control) as well as when all groups are combined (Total). These figures are useful when you need to describe your data.
As you can see, we have 8 participants in the High Dollar Condition, 8 participants in the Low Dollar Condition, and 8 participants in the Control Condition. The mean for the High condition is $337.50 (SD = $37.80), the mean for the Low condition is $293.75 (SD = $34.62), and the mean for the Control condition is $306.25 (SD = $25.88). We can ignore the Std. Error, confidence interval, and minimum maximum for now, but we will need the means and SD information in our write up (below), so we’ll come back to this table.
The One Way ANOVA Table (ANOVA – Useful!)
The One-Way ANOVA table (see ANOVA table below) shows the output of the ANOVA analysis and whether we have a statistically significant difference between our group means. We can see that in this example the significance level is 0.042, which is below 0.05 and, therefore, there is a statistically significant difference in dollar amount recall between two or more of our three conditions. This is great to know, but the ANOVA does not tell us which of the three groups differed. Luckily, we can find this out in the Multiple Comparisons Table and the Homogenous Subsets tables, both of which contain the results of post-hoc tests.
The ANOVA Effect Sizes Table (Not Useful)
The ANOVA Effect Sizes table is a relatively new feature that SPSS provides when you run a One Way ANOVA. This is essentially a power estimation table or an “effect size” table. Consider only the Eta-squared row. This gives you the point estimate, or effect size, which ranges from 0 to 1. The closer to 1 the more powerful the effect. A weak effect would be 0.01, a moderate effect .06, and a strong effect .14 or higher. As you can see here, our effect size of .26 is very strong showing us that our result is strong. However, we will pretty much ignore this table in our write-up. The more important table is the ANOVA table itself (above) and not the “ANOVA Effect Size” table.
Multiple Comparisons Table (Not Useful)
From the results so far we know that there is a significant difference between at least two of our three means. The post hoc table (see below), or Multiple Comparisons table, shows which groups differed from each other. (Note: Hopefully the Tukey post-hoc test is familiar to you, but recognize that there are many different post hoc tests. We will focus exclusively on the Tukey test in this crash course). We can see from the “Multiple Comparisons” table below that there is a significant difference in the dollar amount recalled between the High Dollar Condition and the Low Dollar Condition (p = 0.039). However, there is no difference between the High Dollar Condition and the Control Condition (p = 0.17) and there is no difference between the Low Dollar Condition and the Control Condition (p = 0.73). If you find this table a bit “busy”, you and I agree. A lot of values are duplicated in this table. That’s why I prefer to use the Homogenous Subsets table instead.
Homogenous Subsets Table (Useful)
Like the Multiple Comparisons Table, the Homogenous Subsets table uses Tukey to look at the differences between groups. However, the Homogenous Subsets table separates the conditions by comparing the groups and seeing if the mean for each group falls inside the same versus different subsets. If you look at the table below, you will see that there are two subsets in this table (Subset 1 and Subset 2). Focus on Subset 1 right now. Here, both the Low Dollar Condition (M = 293.75) and the Control Condition (M = 306.25) fall within that same subset. This means that the two conditions do not differ from each other. Similarly, looking at Subset 2, the Control Condition (M = 306.25) and the High Dollar Condition (M = 337.50) do not differ from each other, as both are in the same subset. The important comparison in this output is between Subset 1 and Subset 2. Here, the Low Dollar Condition and the High Dollar Condition are in different subsets, which means that they differ from each other. This table is a bit easier to understand as long as you recognize that when means fall in the same subset, they do not significantly differ from each other but when they fall in different subsets, they do differ significantly from each other.
Below are some examples of the Homogenous Subsets tables in which all means fall in the same Subset 1 (and thus no condition differs from the others, and the ANOVA is likely not significant either) or all means fall in different Subsets 1, 2, and 3 (and thus each condition differs from all of the other conditions):
Reporting the output of the One Way ANOVA
We report the statistics in this format: F(degrees of freedom[df]) = F-value, p = significance level. In our case this would be: F(2, 21) = 3.70, p = 0.042, and our means/SDs would be (M = 337.50, SD = 37.80) for the High Dollar condition, (M = 293.75, SD = 34.62)for the Low Dollar Condition, and (M = 306.25, SD = 25.88) for the Control condition. Just recall that only the High and Low Dollar Conditions differed from each other; neither differed from the Control Condition. We would report the results of the study as follows:
We ran a One Way ANOVA with Dollar Condition (High vs Low vs Control) as our independent variable and the amount of money participants recalled seeing as our dependent variable. The One Way ANOVA was significant, F(2, 21) = 3.70, p = .042. Tukey post hoc tests revealed that participants recalled spending significantly more money on textbooks in the High Dollar Condition (M = $337.50, SD = $37.80) than participants in the Low Dollar Condition (M = $293.75, SD = $34.62). However, Control Condition participants (M = $306.25, SD = $25.88) did not differ in their recall from either High or Low Dollar Condition participants.
That’s it! Not too hard, right? Note that I provided means and standard deviations for each of our three conditions – Thus I expect to see means for all conditions in your papers as well! Also note that p = .042. We no longer use p < .05. The only time we use < is when our p value is .000 or less. In that rare instance, we use p < .001. Otherwise use the equal sign (=).
Remember some other basics here: we use a One Way ANOVA to look at the differences between three or more means to see if the means differ significantly. If they do not differ, then there is no need for post hoc tests. If they do differ, we need post hoc tests. Thus, we need three SPSS tables for a significant ANOVA: the descriptive statistics tables, the One Way ANOVA table, and the post hoc table.
So what does the write-up look like for a non-significant ANOVA? Let’s see:
We ran a One Way ANOVA with Dollar Condition (High vs Low vs Control) as our independent variable and the amount of money participants recalled seeing as our dependent variable. The One Way ANOVA was not significant, F(2, 21) = 1.70, p = .152. Participants recalled spending similar amounts of money on textbooks in the High Dollar Condition (M = $337.50, SD = $37.80), the Low Dollar Condition (M = $293.75, SD = $34.62), and the Control Condition (M = $306.25, SD = $25.88).
Finally, I wanted to show you the means plot (below), just to give you a visual idea about the results for our significant ANOVA
Means Plot – Dollar Condition (IV) and How Much Money Participants Recalled Spending (DV)
Crash Course In Statistics – One Way ANOVA – Quiz #3 (Coaster, Summer 2022)
Instructions: Recall the “excitation-transfer theory” study from your t-Test crash course quiz. Here, we will add another condition to that study so that we can use a One-Way ANOVA.
Do you enjoy rollercoasters? Do you enjoy that rush that you feel when your coaster rushes down that first hill? It can be exhilarating, really getting your heart rate going! But have you noticed that the rollercoaster “high” lasts a long time, even after you get off the ride? How might that “high” impact your responses to other stimuli after your rollercoaster ride?
According to excitation-transfer theory, when a person becomes aroused physiologically, there is a subsequent period of time when the person will continue to experience a high state of residual arousal yet be unaware of it. If additional stimuli are encountered during this time, the individual may mistakenly attribute their residual response from the previous stimuli to future stimuli.
Imagine we run a study to further test excitation-transfer theory. You go to an amusement park to recruit male participants.
For all participants, you show them the dating profile of a moderately attractive young woman and ask them to rate the woman’s dating desirability. That is, you ask “
If you were single, how much would you like to ask her on a date?
” using the scale 1 (not at all) to 7 (very much). But the timing and location of your request differs across conditions. In one condition, you ask men who are standing in line for the rollercoaster to complete the questionnaire. In a second condition, you ask men immediately after they rode the rollercoaster. But you wonder whether standing in line for a rollercoaster may also increase physiological arousal as the men anticipate the rollercoaster. You thus add a third condition where you ask men waiting in line for food at an amusement park restaurant to complete the questionnaire.
Based on excitation-transfer theory, you think that men who just rode a rollercoaster will rate the women higher in dating desirability than men who are waiting in line for the rollercoaster.
That is, the thrill (and adrenaline-based arousal) of having just ridden the rollercoaster should transfer to the woman in the dating profile, leading men to find her more date-worthy after riding the rollercoaster compared to men who are waiting in line to ride the rollercoaster as well as men who are waiting in line for food. You also suspect that men waiting in line for the rollercoaster will find the woman more date-worthy than men standing in line for food.
Complete the questions below and then transfer those answers to your Crash Course in Statistics – The One-Way ANOVA Quiz #3 in Canvas (1 point per question). IMPORTANT: The answer options in Canvas may not be in the same order you see them below, so make sure to copy over the CONTENT of the answer and not simply the answer letter (A, B, C, D, or E). Note: If you want to run these analyses yourself, look for the SPSS file called “#3 ANOVA Crash Course Data Coaster Summer” in Canvas – not required, but definitely recommended!)
1). What is the independent variable in this study?
A. How scary male participants found rollercoasters
(1 = Not at all to 7 = Very much)
B.
How much they would like to ask a woman on a date
(1 = Not at all to 7 = Very much)
C. Whether they had just finished riding a rollercoaster, were waiting to ride a rollercoaster, or were waiting for food at an amusement park restaurant.
D.
Whether they had just finished riding a rollercoaster or were waiting to ride a rollercoaster
E. There is too little information in this study to determine the independent variable.
2). What is the dependent variable in this study?
A. How scary male participants found rollercoasters (1 = Not at all to 7 = Very much)
B. How much they would like to ask a woman on a date (1 = Not at all to 7 = Very much)
C. Whether they had just finished riding a rollercoaster, were waiting to ride a rollercoaster, or were waiting for food at an amusement park restaurant. D. Whether they had just finished riding a rollercoaster or were waiting to ride a rollercoaster
E. There is too little information in this study to determine the dependent variable.
You run a One Way AVOVA on this data set and get the following SPSS output: (Note: The ANOVA Effect Sizes Table is omitted)
3). What are the correct means and standard deviations for the conditions in this study? Round to two decimal places
A. Just rode (M = 4.87, SD = 0.73); Waiting to ride (M = 3.80, SD = 0.85); Waiting for food (M = 4.33, SD = 0.88)
B. Just rode (M = 4.87, SD = 0.88); Waiting to ride (M = 4.33, SD = 0.85); Waiting for food (M = 3.80, SD = 0.73)
C. Just rode (M = 4.87, SD = 0.73); Waiting to ride (M = 4.33, SD = 0.88); Waiting for food (M = 3.80, SD = 0.85)
D. Just rode (M = 3.80, SD = 0.73); Waiting to ride (M = 4.87, SD = 0.88); Waiting for food (M = 4.33, SD = 0.85)
E. Just rode (M = 4.87, SD = 0.13); Waiting to ride (M = 4.33, SD = 0.16); Waiting for food (M = 3.80, SD = 0.15)
4). Is the One-Way ANOVA significant?
A. It is not significant F(2, 89) = 12.60, p = .10
B. It is not significant, F(2, 87) = 8.53, p = .10
C. It is significant, F(2, 89) = 8.53, p < .001
D. It is significant, F(2, 87) = 12.60, p < .001
E. It is significant, F(3, 87) = 12.60, p = .000
5). Finally, which of the following would you use to write out the results in an APA formatted results section? Note that this one is tricky – some answer options differ in only a single number or word! Pay close attention to details here.
A.
We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting for food) as our independent variable and ratings of “If you were single, how much would you like to ask her on a date?”
as our dependent variable. The One Way ANOVA was not significant, F(2, 87) = 12.60, p = .100. Participants said they would like to ask her on a date similarly in the just rode condition (M = 4.87, SD = 0.73), the waiting to ride condition (M = 4.33, SD = 0.88), and the waiting for food condition (M = 3.80, SD = 0.85).
B. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting for food) as our independent variable and ratings of “If you were single, how much would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was significant, F(2, 89) = 12.60, p = .000. Tukey post hoc tests showed that participants said they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73) than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her on a date more in the waiting to ride condition than in the waiting for food condition.
C. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting for food) as our independent variable and ratings of “If you were single, how much would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73) than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food condition (M = 3.80, SD = 0.85), though ratings did not differ between the waiting to ride and waiting for food conditions.
D. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting for food) as our independent variable and ratings of “If you were single, how much would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73) than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her on a date more in the waiting to ride condition than in the waiting for food condition.
E. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting for food) as our independent variable and ratings of “If you were single, how much would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said they would like to ask her on a date more in the waiting for food condition (M = 4.87, SD = 0.73) than in both the just rode condition (M = 4.33, SD = 0.88) and the waiting to ride condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her on a date more in the waiting to ride condition than in the just rode condition.
The One Way ANOVA
So you took Stats I and Stats II passed. But do you remember what you did, how you did it, and
why you did it? If you need some basic statistic reminders for the One Way ANOVA, then this is
the lecture for you! I am going to talk about a One-Way ANOVA example in this document that
corresponds to the same example you saw in the Descriptive Statistics Crash Course (#1) and tTest Crash Course (#2) where participants were asked to recall how much money they spent on
textbooks the prior semester. However, for this ANOVA crash course, we are going to add a third
condition: control (a third group of participants who do not see any prior book recall amounts on
the list). As you can see here, we have ONE independent variable (hence the One Way ANOVA),
but here we have three levels (or three conditions): High, Low, and Control. The good news is that
this mini-lecture will sum up the basics of the ANOVA for you as we look at this study, but you
can find additional information about the ANOVA in your textbooks. On the final pages of this
document are several questions based on this crash course. Answer these questions, and then go
into your “Crash Course in Statistics – The One-Way ANOVA Quiz #3” in your Canvas
assessments menu and copy over your answer. Each Crash Course Quiz counts 5 points.
How, when, and why do a One Way ANOVA?
Before we get to the example, let me give you some basic information about the One Way
ANOVA. Do you recall the t-Test, where we compared two means to see whether and in what
direction the means differed? Well, a One Way ANOVA is very similar, but here we compare
three or more means to see if they differ significantly from one another. In this analysis, we
need three pieces of information: 1) the means for each of the three groups (descriptive
statistics), 2) the One-Way ANOVA information itself, and 3) post hoc tests.
1). Once again, remember that a mean is the average score for that condition. That is, you add
up all of the scores in a condition and divide by the number of total scores to arrive at the
average. Since a One Way ANOVA looks at three or more different conditions, we have at
least three means: one for each condition. The means here (plus the standard deviation, which
we will talk about in the lecture) are descriptive statistics. That is, they help describe the data.
2). The One-Way ANOVA information itself is a test of inferential statistics. That is, we infer
significant differences between the three or more groups. When writing it out, you will see a
very common layout for the One Way ANOVA, something like: F(2, 134) = 2.61, p = .021.
The F tells you this is a One Way ANOVA. The 2 and 134 tells us our degrees of freedom
(more on that in out lecture). The 2.61 is the actual number for the One Way ANOVA. The p
indicates whether it is significant (if it is less than .05, then it is significant).
3). Finally, we have to consider post hoc tests. You might recall using the Tukey post hoc test
in the past, but do you remember why you used it? Take a step back and think about the t-Test,
which looked at two means: Mean A and Mean B. If Mean A is 4.56 and Mean B is 7.67 and
your t-Test is significant (that is, p is less than .05), then you simply compare the two means to
see which is higher: Mean A or Mean B. Here, Mean B is clearly higher (7.67 is higher than
4.56), and since the t-Test is significant then Mean B is significantly higher than Mean A. But
when we have three levels to our independent variable, we are now dealing with three means:
Mean A, Mean B, and Mean C. Let’s say Mean A is 4.56, Mean B is 7.67, and Mean C is 6.21.
If our One Way ANOVA is significant (that is, it is less than .05), we know the means differ.
The question is, which of the three means differ? Does Mean A differ from Mean B? Does
Mean B differ from Mean C? Does Mean A differ from Mean C? Or there might be other
combinations. Maybe Mean A and Mean C do not differ from each other, but both are
significantly lower than Mean B. Unlike the t-Test, we don’t know which of the three means
differ, which is why we run a post hoc test (like Tukey) to compare Mean A to Mean B, and
Mean A to Mean C, and Mean B to Mean C. It runs all of those analyses for us in one test. So
you might wonder, “Why not just run three t-Tests, with one t-Test comparing Mean A to
Mean B, a second t-Test comparing Mean B to Mean C, and a third t-Test comparing Mean B
to Mean C.?” Well, you could actually do that, but we run into a Type I error. That is, the more
tests we run, the greater the chance one of them will be significant. If we run three t-Tests, we
open up the chance of one of them being falsely positive. With the One Way ANOVA, we just
run the one test to compare the three means (note that the post hoc tests are still a part of the
One Way ANOVA – it compares the three means under the umbrella of the One-Way ANOVA
test).
Like the t-Test, we run a One Way ANOVA only under certain conditions.
First, our dependent variable (the variable we measure) must be continuous / scaled. That
is, the DV has to be along a scale. For example, it can be an attitude (“On a scale of 1 to 9,
how angry are you?”), a time frame (“How quickly did the salesperson help the customer
on a scale of zero seconds to a thousand seconds?”), or money estimation (“How much do
you recall spending on textbooks last semester?”). We call these interval or ratio scales,
which means we can use a t-Test or ANOVA. We CANNOT run a One Way ANOVA on
categorical data. That is, if we have a yes / no question (“Are you lonely: Yes or No”) or a
category-based question (“What is your favorite food: hamburgers, pizza, salad, or
tacos?”), then we cannot run a One Way ANOVA. These latter questions are based more
on choice of option rather than an actual rating scale, and thus we cannot use a One Way
ANOVA on them.
Second, we run a One Way ANOVA when we have only one independent variable and that
independent variable has at least three conditions (Note: it can have more than three levels,
but you still only have one independent variable). That is, we compare the means from
Condition A, Condition B, and Condition C. If our One Way ANOVA is significant (p is
less than .05), then we look at our post hoc tests to see which means differ. “The One Way
ANOVA was significant, F(2, 134) = 2.61, p = .021. Tukey post hoc tests showed that
Condition A (mean = 48.38) was significantly lower than Condition B (mean = 63.25). In
addition, Condition C (mean = 48.88) was significantly lower than Condition B. However,
Condition A did not differ significantly from Condition C.”
Let’s see how this looks using the textbook money example.
Textbook Study – How Much Did You Spend On Textbooks (High, Low, or Control)
Recall the basic set-up for our money spent on textbooks. Researchers ask participants to recall
how much they spent on textbooks the prior semester, and has each participant write their
answer on a survey sheet. In two conditions, the first ten answer slots are already filled in,
presumably by other respondents. However, the researcher actually completed those ten slots,
and manipulated the dollar amounts so that in in the High Dollar Condition, the dollar amounts
ranged from $350 to $450 (Figure 1). In the Low Dollar Condition, amounts ranged from $250
to $350 (Figure 2). In our new study, the researcher provides a third condition in which there
are no prior dollar amounts on the list (Control Condition – Figure 3). Using psychological
principles based on conformity and informational social influence (e.g. participants relying on
the behavior of other individuals when they lack a clear memory), the researcher predicts that
those in the High Dollar Condition will recall spending more money on textbooks the prior
semester than those in the Low Dollar Condition, with those in the Control condition providing
a dollar amount somewhere in the middle.
Here, the independent variable is Dollar Condition (High versus Low versus Control) while the
dependent variable is the amount of money participants recall spending on textbooks (in $).
Imagine we have eight real participants in the High Dollar Condition, eight real participants in
the Low Dollar Condition, and eight real participants in the Control Condition (and no, we are
not including the original researcher-completed dollar amounts on the sheet passed out by the
researcher in the High and Low Dollar Conditions, as those are not real participants!).
Figure 3: No Prior Dollar Amount Condition
Consider the data:
Condition A (High)
350
400
375
350
300
325
300
300
∑A = 2700
Mean = $337.50
Condition B (Low)
275
350
325
275
250
260
300
315
∑B = 2350
Mean = $293.75
Condition C (Control)
300
325
300
300
275
350
325
275
∑C = 2450
M = $306.25
∑, or the symbol for Sigma, means “the sum of”. Thus ∑A is the sum of the scores for
Condition A. That is,
+ 400 + 375 + 350 +
+
+
+
= 2700. There are eight
scores here, so we divide 2700 / 8 = 337.50, giving us our mean of $337.50 for Condition A
(High Dollar Condition). We do the same thing for Condition B (Low Dollar Condition),
giving us a mean of $293.75 (2350 / 8 = 293.75). Finally, we do the same thing for Condition
C (Control), giving us a mean of $306.25 (2450 / 8 = 306.25).
For the first part of our analysis, we compare the means. As you see, $293.75 in Condition B
(Low) looks lower than $337.50 in Condition A (High). Now consider the $306.25 in
Condition C (Control), which falls between the High and Low Dollar conditions. That $306.25
doesn’t look that different from the $337.50 in the High Dollar Condition or the $293.75 in the
Low Dollar Condition. If I were “eyeballing” this, I would think that participants recall
spending significantly more in the High Dollar Condition than in the Low Dollar Condition,
but that the Control condition doesn’t differ from either the High or Low Dollar Conditions.
However, just because some of our means seem to differ doesn’t mean they do differ. To make
that assessment, we run the One Way ANOVA and look at the p value to see if it is p is less
than .05. We can do this by hand (like you did in Stats I and Methods One) or we can take the
easy road and let SPSS calculate it for us. I am going to take the easy road, but keep in mind
that we still have to interpret what SPSS tells us.
For the next section, I am going to open SPSS and run a One Way ANOVA. I’ll use
screenshots from SPSS as I go, but feel free to run these analyses yourself. Just set up your
SPSS file like mine (I also included this SPSS file for you in Canvas if you prefer to use that. It
is called “Crash Course Quiz #3 – Textbook Money (ANOVA Practice)”, but it is a short data
set, so I recommend setting up your own SPSS file using the values from the table above). I am
just going to give you the basics here, but you can refer to other sources to figure out some of
the info we get from the One Way ANOVA not covered in this lecture (like homogeneity of
variance, Welch test, etc.).
SPSS – Our Textbook Money Recall Study
1. Click Analyze > Compare Means > One-Way ANOVA … on the top menu.
You will be presented with the following:
2. Put the “Condition (1 = High, 2 = Low, 3 = Control)” variable into the “Factor” box and
the “How much participants recall spending” variable into the “Dependent List” box by
highlighting the relevant variables and pressing the
buttons. Note that SPSS uses
different names for variables. It calls the dependent variable the “Dependent List” and it
calls the independent variable the “Factor”. Just remember that our dependent variable
(Factor) must be scaled in order to run this test (1 to 9, or 1 to 5, or even 0 to 100,000). The
independent variable must be categorical (dressy v. sloppy v. casual, old v. middle aged v.
young, republican v. democrat v. independent, high v. medium v. low, etc.).
3. Click the
Click the
button and tick the “Tukey” checkbox as shown below:
button.
4. Click the
button. Tick the “Descriptive”, and “Means Plot” checkboxes in the
Statistics area as shown below:
Click the
button. Then click the
button.
Output of the One Way ANOVA in SPSS
You will be presented with several tables containing all the data generated by the One-Way
ANOVA procedure in SPSS. Some are useful while others are … less useful (at least to us. They
may be more important if you ever submit an article to a journal, but we will ignore them)!
Descriptive Statistics Table (Useful!)
The descriptives table (see below) provides some very useful descriptive statistics including the
mean, standard deviation and 95% confidence intervals for the dependent variable (How much
participants recalled spending on textbooks) for each separate group (High Dollar Condition, Low
Dollar Condition & Control) as well as when all groups are combined (Total). These figures are
useful when you need to describe your data.
As you can see, we have 8 participants in the High Dollar Condition, 8 participants in the Low
Dollar Condition, and 8 participants in the Control Condition. The mean for the High condition is
$337.50 (SD = $37.80), the mean for the Low condition is $293.75 (SD = $34.62), and the mean
for the Control condition is $306.25 (SD = $25.88). We can ignore the Std. Error, confidence
interval, and minimum maximum for now, but we will need the means and SD information in our
write up (below), so we’ll come back to this table.
The One Way ANOVA Table (ANOVA – Useful!)
The One-Way ANOVA table (see ANOVA table below) shows the output of the ANOVA analysis
and whether we have a statistically significant difference between our group means. We can see
that in this example the significance level is 0.042, which is below 0.05 and, therefore, there is a
statistically significant difference in dollar amount recall between two or more of our three
conditions. This is great to know, but the ANOVA does not tell us which of the three groups
differed. Luckily, we can find this out in the Multiple Comparisons Table and the Homogenous
Subsets tables, both of which contain the results of post-hoc tests.
The ANOVA Effect Sizes Table (Not Useful)
The ANOVA Effect Sizes table is a relatively new feature that SPSS provides when you run a One
Way ANOVA. This is essentially a power estimation table or an “effect size” table. Consider only
the Eta-squared row. This gives you the point estimate, or effect size, which ranges from 0 to 1.
The closer to 1 the more powerful the effect. A weak effect would be 0.01, a moderate effect .06,
and a strong effect .14 or higher. As you can see here, our effect size of .26 is very strong showing
us that our result is strong. However, we will pretty much ignore this table in our write-up. The
more important table is the ANOVA table itself (above) and not the “ANOVA Effect Size” table.
Multiple Comparisons Table (Not Useful)
From the results so far we know that there is a significant difference between at least two of our
three means. The post hoc table (see below), or Multiple Comparisons table, shows which groups
differed from each other. (Note: Hopefully the Tukey post-hoc test is familiar to you, but recognize
that there are many different post hoc tests. We will focus exclusively on the Tukey test in this
crash course). We can see from the “Multiple Comparisons” table below that there is a significant
difference in the dollar amount recalled between the High Dollar Condition and the Low Dollar
Condition (p = 0.039). However, there is no difference between the High Dollar Condition and the
Control Condition (p = 0.17) and there is no difference between the Low Dollar Condition and the
Control Condition (p = 0.73). If you find this table a bit “busy”, you and I agree. A lot of values
are duplicated in this table. That’s why I prefer to use the Homogenous Subsets table instead.
Homogenous Subsets Table (Useful)
Like the Multiple Comparisons Table, the Homogenous Subsets table uses Tukey to look at the
differences between groups. However, the Homogenous Subsets table separates the conditions by
comparing the groups and seeing if the mean for each group falls inside the same versus different
subsets. If you look at the table below, you will see that there are two subsets in this table (Subset
1 and Subset 2). Focus on Subset 1 right now. Here, both the Low Dollar Condition (M = 293.75)
and the Control Condition (M = 306.25) fall within that same subset. This means that the two
conditions do not differ from each other. Similarly, looking at Subset 2, the Control Condition (M
= 306.25) and the High Dollar Condition (M = 337.50) do not differ from each other, as both are in
the same subset. The important comparison in this output is between Subset 1 and Subset 2. Here,
the Low Dollar Condition and the High Dollar Condition are in different subsets, which means that
they differ from each other. This table is a bit easier to understand as long as you recognize that
when means fall in the same subset, they do not significantly differ from each other but when they
fall in different subsets, they do differ significantly from each other.
Below are some examples of the Homogenous Subsets tables in which all means fall in the same
Subset 1 (and thus no condition differs from the others, and the ANOVA is likely not significant
either) or all means fall in different Subsets 1, 2, and 3 (and thus each condition differs from all of
the other conditions):
Reporting the output of the One Way ANOVA
We report the statistics in this format: F(degrees of freedom[df]) = F-value, p = significance level.
In our case this would be: F(2, 21) = 3.70, p = 0.042, and our means/SDs would be (M = 337.50,
SD = 37.80) for the High Dollar condition, (M = 293.75, SD = 34.62) for the Low Dollar
Condition, and (M = 306.25, SD = 25.88) for the Control condition. Just recall that only the High
and Low Dollar Conditions differed from each other; neither differed from the Control Condition.
We would report the results of the study as follows:
We ran a One Way ANOVA with Dollar Condition (High vs Low vs Control) as our
independent variable and the amount of money participants recalled seeing as our
dependent variable. The One Way ANOVA was significant, F(2, 21) = 3.70, p =
.042. Tukey post hoc tests revealed that participants recalled spending significantly
more money on textbooks in the High Dollar Condition (M = $337.50, SD = $37.80)
than participants in the Low Dollar Condition (M = $293.75, SD = $34.62).
However, Control Condition participants (M = $306.25, SD = $25.88) did not differ
in their recall from either High or Low Dollar Condition participants.
That’s it! Not too hard, right? Note that I provided means and standard deviations for each of our
three conditions – Thus I expect to see means for all conditions in your papers as well! Also note
that p = .042. We no longer use p < .05. The only time we use < is when our p value is .000 or less.
In that rare instance, we use p < .001. Otherwise use the equal sign (=).
Remember some other basics here: we use a One Way ANOVA to look at the differences between
three or more means to see if the means differ significantly. If they do not differ, then there is no
need for post hoc tests. If they do differ, we need post hoc tests. Thus, we need three SPSS tables
for a significant ANOVA: the descriptive statistics tables, the One Way ANOVA table, and the
post hoc table.
So what does the write-up look like for a non-significant ANOVA? Let’s see:
We ran a One Way ANOVA with Dollar Condition (High vs Low vs Control) as our
independent variable and the amount of money participants recalled seeing as our
dependent variable. The One Way ANOVA was not significant, F(2, 21) = 1.70, p =
.152. Participants recalled spending similar amounts of money on textbooks in the
High Dollar Condition (M = $337.50, SD = $37.80), the Low Dollar Condition (M =
$293.75, SD = $34.62), and the Control Condition (M = $306.25, SD = $25.88).
Finally, I wanted to show you the means plot (below), just to give you a visual idea about the
results for our significant ANOVA
Means Plot – Dollar Condition (IV) and How Much Money Participants Recalled Spending (DV)
Crash Course In Statistics – One Way ANOVA – Quiz #3 (Coaster, Summer 2022)
Instructions: Recall the “excitation-transfer theory” study from your t-Test crash course quiz.
Here, we will add another condition to that study so that we can use a One-Way ANOVA.
Do you enjoy rollercoasters? Do you enjoy that rush that you feel when your coaster rushes down
that first hill? It can be exhilarating, really getting your heart rate going! But have you noticed that
the rollercoaster “high” lasts a long time, even after you get off the ride? How might that “high”
impact your responses to other stimuli after your rollercoaster ride?
According to excitation-transfer theory, when a person becomes aroused physiologically, there is a
subsequent period of time when the person will continue to experience a high state of residual
arousal yet be unaware of it. If additional stimuli are encountered during this time, the individual
may mistakenly attribute their residual response from the previous stimuli to future stimuli.
Imagine we run a study to further test excitation-transfer theory. You go to an amusement park to
recruit male participants. For all participants, you show them the dating profile of a moderately
attractive young woman and ask them to rate the woman’s dating desirability. That is, you ask “If
you were single, how much would you like to ask her on a date?” using the scale 1 (not at all) to 7
(very much). But the timing and location of your request differs across conditions. In one
condition, you ask men who are standing in line for the rollercoaster to complete the questionnaire.
In a second condition, you ask men immediately after they rode the rollercoaster. But you wonder
whether standing in line for a rollercoaster may also increase physiological arousal as the men
anticipate the rollercoaster. You thus add a third condition where you ask men waiting in line for
food at an amusement park restaurant to complete the questionnaire.
Based on excitation-transfer theory, you think that men who just rode a rollercoaster will rate the
women higher in dating desirability than men who are waiting in line for the rollercoaster. That is,
the thrill (and adrenaline-based arousal) of having just ridden the rollercoaster should transfer to
the woman in the dating profile, leading men to find her more date-worthy after riding the
rollercoaster compared to men who are waiting in line to ride the rollercoaster as well as men who
are waiting in line for food. You also suspect that men waiting in line for the rollercoaster will find
the woman more date-worthy than men standing in line for food.
Complete the questions below and then transfer those answers to your Crash Course in Statistics –
The One-Way ANOVA Quiz #3 in Canvas (1 point per question). IMPORTANT: The answer
options in Canvas may not be in the same order you see them below, so make sure to copy over the
CONTENT of the answer and not simply the answer letter (A, B, C, D, or E). Note: If you want to
run these analyses yourself, look for the SPSS file called “#3 ANOVA Crash Course Data Coaster
Summer” in Canvas – not required, but definitely recommended!)
1). What is the independent variable in this study?
A. How scary male participants found rollercoasters (1 = Not at all to 7 = Very much)
B. How much they would like to ask a woman on a date (1 = Not at all to 7 = Very much)
C. Whether they had just finished riding a rollercoaster, were waiting to ride a rollercoaster, or
were waiting for food at an amusement park restaurant.
D. Whether they had just finished riding a rollercoaster or were waiting to ride a rollercoaster
E. There is too little information in this study to determine the independent variable.
2). What is the dependent variable in this study?
A. How scary male participants found rollercoasters (1 = Not at all to 7 = Very much)
B. How much they would like to ask a woman on a date (1 = Not at all to 7 = Very much)
C. Whether they had just finished riding a rollercoaster, were waiting to ride a rollercoaster, or
were waiting for food at an amusement park restaurant.
D. Whether they had just finished riding a rollercoaster or were waiting to ride a rollercoaster
E. There is too little information in this study to determine the dependent variable.
You run a One Way AVOVA on this data set and get the following SPSS output: (Note: The
ANOVA Effect Sizes Table is omitted)
3). What are the correct means and standard deviations for the conditions in this study?
Round to two decimal places
A. Just rode (M = 4.87, SD = 0.73); Waiting to ride (M = 3.80, SD = 0.85); Waiting for food (M
= 4.33, SD = 0.88)
B. Just rode (M = 4.87, SD = 0.88); Waiting to ride (M = 4.33, SD = 0.85); Waiting for food (M
= 3.80, SD = 0.73)
C. Just rode (M = 4.87, SD = 0.73); Waiting to ride (M = 4.33, SD = 0.88); Waiting for food (M
= 3.80, SD = 0.85)
D. Just rode (M = 3.80, SD = 0.73); Waiting to ride (M = 4.87, SD = 0.88); Waiting for food (M
= 4.33, SD = 0.85)
E. Just rode (M = 4.87, SD = 0.13); Waiting to ride (M = 4.33, SD = 0.16); Waiting for food (M
= 3.80, SD = 0.15)
4). Is the One-Way ANOVA significant?
A. It is not significant F(2, 89) = 12.60, p = .10
B. It is not significant, F(2, 87) = 8.53, p = .10
C. It is significant, F(2, 89) = 8.53, p < .001
D. It is significant, F(2, 87) = 12.60, p < .001
E. It is significant, F(3, 87) = 12.60, p = .000
5). Finally, which of the following would you use to write out the results in an APA formatted
results section? Note that this one is tricky – some answer options differ in only a single
number or word! Pay close attention to details here.
A. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs.
Waiting for food) as our independent variable and ratings of “If you were single, how much
would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was
not significant, F(2, 87) = 12.60, p = .100. Participants said they would like to ask her on a
date similarly in the just rode condition (M = 4.87, SD = 0.73), the waiting to ride condition
(M = 4.33, SD = 0.88), and the waiting for food condition (M = 3.80, SD = 0.85).
B. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs.
Waiting for food) as our independent variable and ratings of “If you were single, how much
would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was
significant, F(2, 89) = 12.60, p = .000. Tukey post hoc tests showed that participants said
they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73)
than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food
condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her
on a date more in the waiting to ride condition than in the waiting for food condition.
C. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs.
Waiting for food) as our independent variable and ratings of “If you were single, how much
would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was
significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said
they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73)
than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food
condition (M = 3.80, SD = 0.85), though ratings did not differ between the waiting to ride and
waiting for food conditions.
D. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs.
Waiting for food) as our independent variable and ratings of “If you were single, how much
would you like to ask her on a date?” as our dependent variable. The One Way ANOVA was
significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said
they would like to ask her on a date more in the just rode condition (M = 4.87, SD = 0.73)
than in both the waiting to ride condition (M = 4.33, SD = 0.88) and the waiting for food
condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her
on a date more in the waiting to ride condition than in the waiting for food condition.
E. We ran a One-Way ANOVA with coaster condition (Just rode vs. Waiting to ride vs. Waiting
for food) as our independent variable and ratings of “If you were single, how much would
you like to ask her on a date?” as our dependent variable. The One Way ANOVA was
significant, F(2, 87) = 12.60, p < .001. Tukey post hoc tests showed that participants said
they would like to ask her on a date more in the waiting for food condition (M = 4.87, SD =
0.73) than in both the just rode condition (M = 4.33, SD = 0.88) and the waiting to ride
condition (M = 3.80, SD = 0.85). In addition, participants also said they would like to ask her
on a date more in the waiting to ride condition than in the just rode condition.
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