The syntax file for this seminar.

There are a number of situations that can arise when the analysis includes between groups effects as well as within subject effects. We start by showing 4 example analyses using measurements of depression over 3 time points broken down by 2 treatment groups. In the first example we see that the two groups differ in depression but neither group changes over time. In the second example the two groups grow in depression but at the same rate over time. In the third example, the two groups start off being quite different in depression but over time the groups get closer in their level of depression. The fourth example shows the groups starting off at the same level of depression, and one group group increases over time whereas the other group decreases over time.

Note that in the interest of making learning the concepts easier we have taken the liberty of using only a very small portion of the voluminous output that SPSS provides and we have inserted the graphs in the beginning of the output rather than at the end.

**Demo Analysis #1**

The between groups test indicates that there the variable **group** is significant, consequently in the graph we see that
the lines for the two groups are rather far apart. The within subject test indicate that there is not a
significant **time** effect, in other words, the groups do not change
in depression over
time. In the graph we see that the groups have lines that are flat, i.e. the
slopes of the lines are approximately equal to zero.
Also, since the lines are parallel, we are not surprised that there is no
interaction.

<Abbreviated output from GLM command>

**Demo Analysis #2**

The between groups test indicates that there the variable **group** is not significant, consequently in the graph we see that
the lines for the two groups are rather close together. The within subject test indicate that there is a
significant **time** effect, in other words, the groups do change
in depression over time. In the graph we see that the groups have lines that increase over time.
Again, the lines are parallel consistent with the finding
that the interaction is not significant.

<Abbreviated output from GLM command>

**Demo Analysis #3**

The between groups test indicates that there the variable **group** is significant, consequently in the graph we see that
the lines for the two groups are rather far apart. The within subject test indicate that there is a
significant **time** effect, in other words, the groups do change over time,
both groups are getting less depressed over time.
Moreover, the interaction of **time** and **group** is significant which means that the
groups are changing over time but are changing in different ways, which means that in the graph the lines will
not be parallel. In the graph
we see that the groups have non-parallel lines that decrease over time and are getting
progressively closer together over time.

<Abbreviated output from GLM command>

**Demo Analysis #4**

The within subject test indicate that the interaction of **time** and **group **is significant. The
main effect of **time **is not significant. However, the significant interaction indicates that
the groups are changing over time and they are changing in
different ways, in other words, in the graph the lines of the groups will not be parallel.
The between groups test indicates that there the variable **group** is
significant. In the graph for this particular case we see that one group is
increasing in depression over time and the other group is decreasing
in depression over time.

<Abbreviated output from GLM command>

**Exercise data examples**

The data consists of people who were randomly assigned to two different diets: low-fat and not low-fat and three different types of exercise: at rest, walking leisurely and running. Their pulse rate was measured at three different time points during their assigned exercise: at 1 minute, 15 minutes and 30 minutes.

data list free / id exertype diet time1 time2 time3. begin data. 1 1 1 85 85 88 2 1 1 90 92 93 3 1 1 97 97 94 4 1 1 80 82 83 5 1 1 91 92 91 6 1 2 83 83 84 7 1 2 87 88 90 8 1 2 92 94 95 9 1 2 97 99 96 10 1 2 100 97 100 11 2 1 86 86 84 12 2 1 93 103 104 13 2 1 90 92 93 14 2 1 95 96 100 15 2 1 89 96 95 16 2 2 84 86 89 17 2 2 103 109 90 18 2 2 92 96 101 19 2 2 97 98 100 20 2 2 102 104 103 21 3 1 93 98 110 22 3 1 98 104 112 23 3 1 98 105 99 24 3 1 87 132 120 25 3 1 94 110 116 26 3 2 95 126 143 27 3 2 100 126 140 28 3 2 103 124 140 29 3 2 94 135 130 30 3 2 99 111 150 end data.

## Exercise example, model 1 (time and diet)

Let us first consider the model including **diet** as the group variable.
The graph below suggests that the pulse rate is growing over time. The
pulse rates may vary for the 2 diets and it is possible that the pulse rate is
growing faster for the “green” diet than the “red” diet.

title "Exercise example model 1, time and diet". GLM time1 time2 time3 BY diet /WSFACTOR = time 3 /WSDESIGN=time /DESIGN=diet /EMMEANS=tables( time*diet) /PLOT = PROFILE( time*diet).

Looking at the results from the **manova **test
the effect of **time** is significant but the interaction of
**time** and **diet** is not significant. The between subject test of the
effect of
**diet** is also not significant. Consequently, in the graph we have lines
that are not flat, in fact, they are actually increasing over time, which was
expected since the effect of time was significant. Furthermore, the lines are
approximately parallel which was anticipated since the interaction was not
significant.

## Exercise example, model 2 (time and exercise type)

Next, let us consider the model including **exertype** as the group variable.

title "Exercise example model 2, time and exertype". GLM time1 time2 time3 BY exertype /WSFACTOR = time 3 /WSDESIGN=time /DESIGN=exertype /EMMEANS=tables( time*exertype) /PLOT = profile( time*exertype).

The interaction of **time** and **exertype** is significant as is the
effect of **time**. The between subject test of the effect of **exertype **is
also significant. Consequently, in the graph we have lines that are not parallel which we expected
since the interaction was significant. Furthermore, we see that some of the lines that are rather far
apart and at least one line is not horizontal which was anticipated since **exertype** and **time**
were both significant. The output for this analysis is
omitted.

## Further Issues

**Missing Data**

- Compare GLM and Mixed on Missing Data

**Variance-Covariance Structures**

- Discuss “univariate” vs. “multivariate” tests.
- Discuss “sphericity” and test of sphericity.

**Independence**

As though analyzed using between subjects analysis.

s

^{2}0 s^{2}0 0 s^{2}

**Compound Symmetry**

The **univariate** tests assumes that the variance-covariance
structure has compound symmetry. There
is a single Variance (represented by s^{2})
for all 3 of the time points
and there is a single covariance (represented by s_{1})
for each of the pairs of trials. This is illustrated
below.

s

^{2}s_{1}s^{2}s_{1}s_{1}s^{2}

**Unstructured**

The **manova** tests assumes that each
variance and covariance is unique, see below, referred to
as an **unstructured**

covariance matrix. Each trial has its
own variance (e.g. s_{1}^{2}
is the variance of trial 1) and each pair of trials has its own
covariance (e.g. s_{21} is the covariance
of trial 1 and trial2).

s

_{1}^{2}s_{21}s_{2}^{2}s_{31}s_{32}s_{3}^{2}

We can use the **sphericity** test to indicate
which is most appropriate: the **manova** or the **univariate** test. The null hypothesis test of the
test of **sphericity** is: the variance-covariance structure has the Huynh-Feldt
structure, so called Type H structure. If the **sphericity** test is not significant then
we can not reject that null hypothesis that the variance-covariance
structure has Type H structure. A Type H structure is a variance-covariance
structure more general than compound symmetry structure that allows the use of the **univariate** tests. If, however, the **sphericity** test is significant then we reject that
the variance-covariance structure has
a Type H structure and it is most appropriate to use the results from the **manova** test or alternatively
use the corrections for the **univariate** test. It is very important, however, to note that the **sphericity **test is overly sensitive. It is very likely to reject compound
symmetry when the data only slightly deviates from compound symmetry,
so in actuality this test could be very deceiving and may be best ignored.

**Autoregressive**

Another common covariance structure which is frequently
observed in repeated measures data is an **autoregressive** structure, which
recognizes that observations which are more proximate are more correlated than
measures that are more distant

s

^{2}sr s^{2}sr^{2}sr s^{2}

**Huynh-Feldt Variances**

This is a circular matrix that satisfies the condition:
s_{i}^{2} + s_{j}^{2}
– s_{ij} = 2l.

s

_{1}^{2}(s

_{1}^{2}+ s_{2}^{2})/2 – l s_{2}^{2}(s_{1}^{2}+ s_{3}^{2})/2 – l (s_{2}^{2}+ s_{3}^{2})/2 – l s_{3}^{2}

However, we cannot use this kind of covariance structure in a traditional repeated measures analysis, but we can use the MIXED command for such an analysis.

For a complete list of all variance-covariance structures
that SPSS supports in the **mixed** command please see refer to the SPSS manual.

Let’s look at the correlations, variances and covariances for the exercise data.

CORRELATIONS /VARIABLES=time1 time2 time3 /STATISTICS XPROD.

## Exercise example, model 2 using MIXED Command

Even though we are very impressed with our results so far, we are not completely convinced that the variance-covariance structure really has compound symmetry. In order to compare models with different variance-covariance structures we have to use proc mixed and try the different structures that we think our data might have. However, in order to use proc mixed we must reshape our data from its wide form to a long form.

VARSTOCASES /MAKE pulse from time1 TO time3 /INDEX = time(3). print / id exertype diet time pulse. execute.

Looking at the first 14 observations of the long dataset.

1 1.00 1.00 1 85.00 1 1.00 1.00 2 85.00 1 1.00 1.00 3 88.00 2 1.00 1.00 1 90.00 2 1.00 1.00 2 92.00 2 1.00 1.00 3 93.00 3 1.00 1.00 1 97.00 3 1.00 1.00 2 97.00 3 1.00 1.00 3 94.00 4 1.00 1.00 1 80.00 4 1.00 1.00 2 82.00 4 1.00 1.00 3 83.00 5 1.00 1.00 1 91.00 5 1.00 1.00 2 92.00

Compound Symmetry

The first model we will look at is one using compound symmetry for the variance-covariance
structure. This model should confirm the results of the **univariate** tests that we obtained through **glm **and we will be
able to obtain fit statistics that we will use for comparisons with our models that assume other
variance-covariance structures.

title "model 2, compound symmetry". MIXED pulse BY exertype time /FIXED = exertype time exertype*time /REPEATED = time | SUBJECT(id) COVTYPE(cs).

**Unstructured**

We now try an unstructured covariance matrix.

title "model 2, unstructured". MIXED pulse BY exertype time /FIXED = exertype time exertype*time /REPEATED = time | SUBJECT(id) COVTYPE(un).

Autoregressive

From previous studies we suspect that our data might actually have an auto-regressive variance-covariance structure so this is the model we will look at next. However, for our data the auto-regressive variance-covariance structure does not fit our data much better than the compound symmetry does (AIC of 594.8 vs. 594.1).

title "model 2, autoregressive". MIXED pulse BY exertype time /FIXED = exertype time exertype*time /REPEATED = time | SUBJECT(id) COVTYPE(ar1).

These results are exactly the same as those from the compound symmetry model. It is very important to explore different variance-covariance structures where we have fit statistics indicating how well the models fit compared to each other.

Model comparison (comparing to Compound Symmetry)

Model |
AIC |
-2RLL |
Parms
(df + 1) |
Diff -2RLL
(vs. CS) |
Diff in df
(vs. CS) |
p value for Diff
(from a chi square
dist) |

Compound Symmetry |
594.8 | 590.8 | 2 | |||

Unstructured |
589.7 | 577.7 | 6 | 13.1 | 4 | .01 |

Autoregressive |
594.1 | 590.1 | 2 | .7 | 0 | na |

(from SAS PROC MIXED)
Autoregressive Heterogenous Variances |
587.8 | 579.8 | 4 | 11 | 2 | 0.027 |

The two most promising structures are **Autoregressive
Heterogeneous Variances** and **Unstructured**.

## Exercise example, model 3 (time, diet and exertype) Using the GLM command

Looking at models including only **diet** or **exertype** separately does not answer all our questions. We would also
like to know if the
people on the low-fat diet who engage in running have lower pulse rates than the people participating
in the not low-fat diet who are not running. In order to address these types of questions we need to look at
a model that includes the interaction of **diet **and **exertype**.
Let us first look at the model including both **diet** and **exertype**
using the **glm** command which means we have to use the wide data again. If we are not satisfied with the choice of
compound symmetry or unstructured variance-covariance structure then we can
return to the **mixed** command.

In the graph of **exertype** by diet we see that for the low-fat diet (**diet**=1) group the pulse rate for the two
exercise types: at rest and walking, are very close together and they are
almost flat whereas the running group has a higher pulse rate that increases over time. For the
not low-fat diet (**diet**=2) group the same two exercise types: at rest and walking, are also very close
together and almost flat. For this group, however, the pulse rate for the running group increases greatly
over time and the increase is much steeper than the increase of the running group in the low-fat diet group.

The test of **sphericity **is not significant so it is appropriate to look at the results from the **univariate **tests. The
**univariate** tests indicate that there is a three-way interaction between
**diet**, **exertype** and **time**. In other words, the pulse rate will depend on which diet you follow, the exercise type
you engage in and at what time during the exercise that you measure the pulse. The interactions of **time **and **exertype** and **diet **and **exertype **are also
significant as are the main effects of **diet **and **exertype**.

title "model 3, time exertype and diet". GLM time1 time2 time3 BY exertype diet /WSFACTOR=time 3 /WSDESIGN=time /PLOT = profile(time*exertype*diet ) .

Looking at the graphs of **exertype** by **diet**.

## Exercise example, model 3 (time, diet and exertype) Using the MIXED command

For the mixed model we will use the autoregressive heterogeneous variances variance-covariance structure
since we previously observed that this is the structure that appears to fit the data the best (see discussion
of variance-covariance structures). We do not expect to find a great change in which factors will be significant
but we do expect to have a model that has a better fit than the glm model.
The graphs are exactly the same as the glm model and we find that the same factors are significant. However, since
the model has a better fit we can be more confident in the estimate of the standard errors and therefore we can
be more confident in the tests and in the findings of significant factors. The model has a better fit than the
model which only includes **exertype** and **time** because both the -2Log Likelihood and the AIC has decrease dramatically.
The -2 Log Likelihood decreased from 579.8 for the model including only **exertype** and
**time** to 505.3
for the current model; the AIC decreased from 587.8 for the model including only
**exertype** and **time**
to 513.3 for the current model.

MIXED pulse BY exertype diet time /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1).

**Contrasts and interaction contrasts for model 3**

From the graphs in the above analysis we see that the runners (**exertype** level 3) have a
pulse rate that is increases much quicker than the pulse rates of the two other groups. We would like
to know if there is a statistically significant difference between the changes over time in the pulse
rate of the runners versus the change over time in the pulse rate of the walkers and the “rest-ers”
(the people at rest) across diet groups and across time. Furthermore, we suspect that there might be
a difference in pulse rate over time and across exercise type between the two diet groups. But to make
matters even more complicated we would like to test if the runners in the low fat diet group are
statistically significantly different from all the other groups (i.e. the runners in the non-low
fat diet, the walkers and the “rest-ers” in both diet groups). Since we are being
ambitious we also want to test if the runners in the low fat diet group (**diet**=1) are different
from the runners in the non-low fat diet group (**diet**=2). These contrasts are all tested
using the **test** subcommand.

In the following example we are testing whether the pulse rate of **exertype** groups 1 and 2 is
significantly different from the pulse rate of **exertype** group 3 (the runners). We are conducting
this test across both **time** and **diet**. The results are contained in the table labeled “Contrast
Estimates” which are the results of the **test** subcommand and here we note that the runners are indeed
statistically significantly different from the other two groups (p < .0001). Since we are conducting a test involving
**exertype** we have to include the coding scheme for all interactions which involve **exertype**. This is
the reason why we have included the coding for the interaction **diet*exertype**, **exertype*time** and
**exertype*time*diet** in the **test** subcommand.
Note: The only change in the results from the model in the above example is the inclusion of the
“Contrast Estimates” table and we will therefore omit them in the subsequent examples.

*exertype 12 v 3. *include all interactions involving exertype.MIXED pulse BY exertype time diet /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /test= 'exer 12 v 3' exertype -1/2 -1/2 1 exertype*diet -1/4 -1/4 -1/4 -1/4 1/2 1/2 exertype*time -1/6 -1/6 -1/6 -1/6 -1/6 -1/6 1/3 1/3 1/3 exertype*time*diet -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 -1/12 1/6 1/6 1/6 1/6 1/6 1/6.

In the following example we are testing if **exertype** group 1 is different from **exertype**
group 2 across both **time** and **diet**. From the results we conclude that they are not
statistically significantly different from each other at the 0.05 level (p = 0.07).

*exertype 1 v 2. *include all interactions involving exertype.MIXED pulse BY exertype time diet /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /test= 'exer 1 v 2' exertype -1 1 0 exertype*diet -1/2 -1/2 1/2 1/2 0 0 exertype*time -1/3 -1/3 -1/3 1/3 1/3 1/3 0 0 0 exertype*time*diet -1/6 -1/6 -1/6 -1/6 -1/6 -1/6 1/6 1/6 1/6 1/6 1/6 1/6 0 0 0 0 0 0.

In the following example we are testing if there is a significant difference
between the two diets across **time** and **exertype**. From the results we conclude
that the two diets are indeed statistically significantly different (p = 0.001).

*diet 1 v 2. *include all interactions involving diet.MIXED pulse BY exertype time diet /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /test= 'diet 1 v 2' diet -1 1 exertype*diet -1/3 1/3 -1/3 1/3 -1/3 1/3 time*diet -1/3 1/3 -1/3 1/3 -1/3 1/3 exertype*time*diet -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9 -1/9 1/9.

In the following example we are testing if there is a difference across diets between the differences
of **exertype** group 1 and 2 versus **exertype** group 3. This kind of contrast is known as an
interaction contrast and for more details about these type of interactions please refer to
Design and Analysis by G. Keppel. From the results we see that there is indeed a significant differences across diets between
**exertype** group 1 and 2 and **exertype** group 3 (p = 0.004).

*diet 1 v 2 & exertype 12 v 3. *include all interactions including exertype*diet.MIXED pulse BY exertype time diet /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /test= 'diet 1 v 2 and exertype 12 v 3' exertype*diet -1/2 1/2 -1/2 1/2 1 -1 exertype*time*diet -1/6 1/6 -1/6 1/6 -1/6 1/6 -1/6 1/6 -1/6 1/6 -1/6 1/6 1/3 -1/3 1/3 -1/3 1/3 -1/3.

In the following example we are testing the difference between the diets for
**exertype** group 3
(the runners). From the results we see that there is a statistically significant difference between
the diets (p < 0.001).

*diet 1 v 2 for runners only (exertype group 3). *include all interactions invovling diet.MIXED pulse BY exertype time diet /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /test= 'diet 1 v 2 runners only' diet 1 -1 exertype*diet 0 0 0 0 1 -1 time*diet 1/3 -1/3 1/3 -1/3 1/3 -1/3 exertype*time*diet 0 0 0 0 0 0 0 0 0 0 0 0 1/3 -1/3 1/3 -1/3 1/3 -1/3.

It is also possible to look at the differences among groups at each level of another variable and
in order to do so we have to utilize the **emmeans** subcommand with the **compare** option.
In the following example, the first **emmeans** subcommand tests for differences among the
**exertype** groups at each level of **diet** across all levels of **time**; the second
**emmeans** subcommand tests for differences in groups of **exertype** for each time point
across both levels of **diet**; the third **emmeans** subcommand tests for differences in groups of
**exertype** for each combination of **time** and **diet** levels.
The results of the first **emmeans** subcommand indicate that there is a differences in the means for
each level of **exertype** when we are considering only the people in **diet
=** 1 (p < 0.001) and when we are
considering only the people in **diet** = 2 (p < 0.001). The results of the second **emmeans**
subcommand indicate that at **time** = 1 there is no differences in the means of the pulse rate for
the different **exertype** groups; at **time** = 2 there is a difference among the means of the
**exertype** groups (p < 0.001); at **time** = 3 there is also a difference among the means
of pulse rate of the **exertype** groups (p < 0.001). The results of the third emmeans subcommand
indicates that for **diet** = 1 and **time** = 1 there is no significant difference across the
**exertype** groups (p = .34); for **diet** = 1 and **time** = 2 there is a significant
difference across the **exertype** groups (p = 0.003); for **diet** = 1 and **time** = 3 there is a
significant difference across the **exertype** groups(p < 0.001). The results of the third **emmeans **
subcommand also indicates that for **diet** = 2 and **time** = 1 there is no significant
difference across the **exertype** groups (p = .229); for **diet** = 2 and **time** = 2 there is a
significant difference across the **exertype** groups (p < 0.001); for **diet** = 2 and **time** = 3
there is a significant difference across the **exertype** groups(p < 0.001). To summarize these
results we could say that at the first time point the three **exertype** groups have mean pulse rates that
are not significantly different regardless of which diet they are following. For the other time points
there are significant differences between the mean pulse rates for the three **exertype** groups and that
these differences exists in both **diet** groups.

*The first emmeans subcommand is testing the differences among exertype for each level of diet. *The second emmeans subcommand is testing for differences at each time point. *The 3rd emmeans subcommand tests for differences in groups of exertype for each combination of time and diet levels.MIXED pulse BY exertype diet time /FIXED = exertype diet time exertype*time diet*time diet*exertype exertype*diet*time /REPEATED = time | SUBJECT(id) COVTYPE(arh1) /emmeans = tables(exertype*diet) compare(exertype) /emmeans = tables(exertype*time) compare(exertype) /emmeans = tables(exertype*diet*time) compare(exertype).

### Unequally Spaced Time Points

## Modeling Time as a Linear Predictor of Pulse

We have another study which is very similar to the one previously discussed except that
in this new study the pulse measurements were not taken at regular time points.
In this study a baseline pulse measurement was obtained at time = 0 for every individual
in the study. However, subsequent pulse measurements were taken at less
regular time intervals. The second pulse measurements were taken at approximately 2 minutes
(120 seconds); the third pulse measurement was obtained at approximately 5 minutes (300 seconds);
and the fourth and final pulse measurement was obtained at approximately 10 minutes (600
seconds). The data for this study is displayed below in the **
study2** data file.

data list free / id timec exertype diet pulse time. begin data. 1 1 1 1 90 0 1 2 1 1 92 228 1 3 1 1 93 296 1 4 1 1 93 639 2 1 1 1 90 0 2 2 1 1 92 56 2 3 1 1 93 434 2 4 1 1 93 538 3 1 1 1 97 0 3 2 1 1 97 150 3 3 1 1 94 295 3 4 1 1 94 541 4 1 1 1 80 0 4 2 1 1 82 121 4 3 1 1 83 256 4 4 1 1 83 575 5 1 1 1 91 0 5 2 1 1 92 161 5 3 1 1 91 252 5 4 1 1 91 526 6 1 1 2 83 0 6 2 1 2 83 73 6 3 1 2 84 320 6 4 1 2 84 570 7 1 1 2 87 0 7 2 1 2 88 40 7 3 1 2 90 325 7 4 1 2 90 730 8 1 1 2 92 0 8 2 1 2 94 205 8 3 1 2 95 276 8 4 1 2 95 761 9 1 1 2 97 0 9 2 1 2 99 57 9 3 1 2 96 244 9 4 1 2 96 695 10 1 1 2 100 0 10 2 1 2 97 143 10 3 1 2 100 296 10 4 1 2 100 722 11 1 2 1 86 0 11 2 2 1 86 83 11 3 2 1 84 262 11 4 2 1 84 566 12 1 2 1 93 0 12 2 2 1 103 116 12 3 2 1 104 357 12 4 2 1 104 479 13 1 2 1 90 0 13 2 2 1 92 191 13 3 2 1 93 280 13 4 2 1 93 709 14 1 2 1 95 0 14 2 2 1 96 112 14 3 2 1 100 219 14 4 2 1 100 367 15 1 2 1 89 0 15 2 2 1 96 96 15 3 2 1 95 339 15 4 2 1 95 639 16 1 2 2 84 0 16 2 2 2 86 92 16 3 2 2 89 351 16 4 2 2 89 508 17 1 2 2 103 0 17 2 2 2 109 196 17 3 2 2 114 213 17 4 2 2 120 634 18 1 2 2 92 0 18 2 2 2 96 117 18 3 2 2 101 227 18 4 2 2 101 614 19 1 2 2 97 0 19 2 2 2 98 70 19 3 2 2 100 295 19 4 2 2 100 515 20 1 2 2 102 0 20 2 2 2 104 165 20 3 2 2 103 302 20 4 2 2 103 792 21 1 3 1 93 0 21 2 3 1 98 100 21 3 3 1 110 396 21 4 3 1 115 498 22 1 3 1 98 0 22 2 3 1 104 104 22 3 3 1 112 310 22 4 3 1 117 518 23 1 3 1 98 0 23 2 3 1 105 148 23 3 3 1 118 208 23 4 3 1 121 677 24 1 3 1 87 0 24 2 3 1 122 171 24 3 3 1 127 320 24 4 3 1 133 633 25 1 3 1 94 0 25 2 3 1 110 57 25 3 3 1 116 268 25 4 3 1 119 657 26 1 3 2 95 0 26 2 3 2 126 163 26 3 3 2 143 382 26 4 3 2 147 501 27 1 3 2 100 0 27 2 3 2 126 70 27 3 3 2 140 347 27 4 3 2 148 737 28 1 3 2 103 0 28 2 3 2 124 61 28 3 3 2 140 263 28 4 3 2 143 588 29 1 3 2 94 0 29 2 3 2 135 164 29 3 3 2 130 353 29 4 3 2 137 560 30 1 3 2 99 0 30 2 3 2 111 114 30 3 3 2 140 362 30 4 3 2 148 501 end data.

In order to get a better understanding of the data we will look at a scatter plot of the data with lines connecting the points for each individual.

This is a situation where multilevel modeling excels for the analysis of data with irregularly spaced time points. The multilevel model with time as a linear effect is illustrated in the following equations.

Level 1 (time): Pulse = β_{0j }+ β_{1j}
(Time) + r_{ij}

Level 2 (person): β_{0j }= γ_{00 }
+ γ_{01}(Exertype) + u_{0j
}Level 2 (person): β_{1j }= γ_{10 } + γ_{11}(Exertype)
+ u_{1j}

Substituting the level 2 model into the level 1 model we get the following single equations. Note: The random components have been placed in square brackets.

Pulse = γ_{00 }+ γ_{01}(Exertype)
+ γ_{10}(Time) + γ_{11}(Exertype*time) + [ u_{0j }+_{
}u_{1j}(Time) + r_{ij} ]

Since this model contains both fixed and random components, it can be
analyzed in the **mixed** command as shown below.

*linear model. mixed pulse with time by exertype /fixed = time exertype time*exertype /random = intercept time | subject(id).

In order to access how well the model with time as a linear effect fits the model
we have plotted the predicted and the observed values in one plot. As we can easily
see the this model is not an optimal model since the green line is fitting curved data
with a straight line. All the predicted lines are straight lines since we fitted
**time** to as a linear effect.

Note: Currently it is not possible to produce these types of graphs with the **
mixed** command in SPSS version 11. The graphs in this section were all produced by SAS.
Please refer to the seminar
Repeated Measures using SAS for the code.

## Modeling Time as a Quadratic Predictor of Pulse

To model the quadratic effect of time, we add **time*time** to
the model. We see that term is significant.

*quadratic mode. mixed pulse with time by exertype /fixed = time exertype time*exertype time*time /random = intercept time | subject(id).

Again, we can plot the predicted values against the actual values of pulse. We see that this model fits better, but it appears that the predicted values for the green group have too little curvature and the red and blue group have too much curvature.

## Modeling Time as a Quadratic Predictor of Pulse, Interacting by Exertype

We can include an interaction of **time*****time*****exertype** to indicate that the
different exercises not only show different linear trends over time, but that
they also show different quadratic trends over time, as shown below. The
**time*****time*****exertype** term is significant.

*Modeling time as quadratic and interacting with exertype. mixed pulse with time by exertype /fixed = time exertype time*exertype time*time time*time*exertype /random = intercept time | subject(id).

Below we see the predicted and actual values and see that this model fits much better. The green curve hugs the data from the green group much better and the blue and red groups are much flatter, fitting their data much better as well.

### For more information:

- SPSS Library: Comparing Methods of Analyzing Repeated Measures Data
- SPSS Textbook Examples: Design and Analysis- A Researchers Handbook (3rd Edition)
- SPSS Library: An Overview of SPSS GLM (courtesy of SPSS)
- SPSS Library: MANOVA and GLM (courtesy of SPSS)
- SPSS Library: Understanding and Interpreting Parameter Estimates in Regression and ANOVA (courtesy of SPSS)
- SPSS Library: How do I handle interactions of continuous and categorical variables?
- SPSS Library: Advanced Issues in Using and Understanding SPSS MANOVA (courtesy of SPSS)