Introduction to SSM Analysis

Circumplex models, scales, and data

Circumplex models are popular within many areas of psychology because they offer a parsimonious account of complex psychological domains, such as emotion and interpersonal functioning. This parsimony is achieved by understanding phenomena in a domain as being a “blend” of two primary dimensions. For instance, circumplex models of emotion typically represent affective phenomena as a blend of valence (pleasantness versus unpleasantness) and arousal (activity versus passivity), whereas circumplex models of interpersonal functioning typically represent interpersonal phenomena as a blend of communion (affiliation versus separation) and agency (dominance versus submissiveness). These models are often depicted as circles around the intersection of the two dimensions (see figure). Any given phenomenon can be located within this circular space through reference to the two underlying dimensions (e.g., anger is a blend of unpleasantness and activity).

Circumplex scales contain multiple subscales that attempt to measure different blends of the two primary dimensions (i.e., different parts of the circle). Although there have historically been circumplex scales with as many as sixteen subscales, it has become most common to use eight subscales: one for each “pole” of the two primary dimensions and one for each “quadrant” that combines the two dimensions. In order for a set of subscales to be considered circumplex, they must exhibit certain properties. Circumplex fit analyses can be used to quantify these properties.

Circumplex data is composed of scores on a set of circumplex scales for one or more participants (e.g., persons or organizations). Such data is usually collected via self-report, informant-report, or observational ratings in order to locate psychological phenomena within the circular space of the circumplex model. For example, a therapist might want to understand the interpersonal problems encountered by an individual patient, a social psychologist might want to understand the emotional experiences of a group of participants during an experiment, and a personality psychologist might want to understand what kind of interpersonal behaviors are associated with a trait (e.g., extraversion).

The Structural Summary Method

The Structural Summary Method (SSM) is a technique for analyzing circumplex data that offers practical and interpretive benefits over alternative techniques. It consists of fitting a cosine curve to the data, which captures the pattern of correlations among scores associated with a circumplex scale (i.e., mean scores on circumplex scales or correlations between circumplex scales and an external measure). By plotting a set of example scores below, we can gain a visual intuition that a cosine curve makes sense in this case. First, we can examine the scores with a bar chart ignoring the circular relationship among them.

Next, we can leverage the fact that these subscales have specific angular displacements in the circumplex model (and that 0 and 360 degrees are the same) to create a path diagram.

This already looks like a cosine curve, and we can finally use the SSM to estimate the parameters of the curve that best fits the observed data. By plotting it alongside the data, we can get a sense of how well the model fits our example data.

Understanding the SSM parameters

The SSM estimates a cosine curve to the data using the following equation: \[S_i = e + a \times \cos(\theta_i - d)\] where \(S_i\) and \(\theta_i\) are the score and angle on scale \(i\), respectively, and \(e\), \(a\), and \(d\) are the elevation, amplitude, and displacement parameters, respectively. Before we discuss these parameters, however, we can also estimate the fit of the SSM model. This is essentially how close the cosine curve is to the observed data points. Deviations (in red, below) will lower model fit.

If fit is less than 0.70, it is considered “unacceptable” and only the elevation parameter should be interpreted. If fit is between 0.70 and 0.80, it is considered “adequate,” and if it is greater than 0.80, it is considered “good.” Sometimes SSM model fit is called prototypicality or denoted using \(R^2\).

The first SSM parameter is elevation or \(e\), which is calculated as the mean of all scores. It is the size of the general factor in the circumplex model and its interpretation varies from scale to scale. For measures of interpersonal problems, it is interpreted as generalized interpersonal distress. When using correlation-based SSM, \(|e|\ge.15\) is considered “marked” and \(|e|<.15\) is considered “modest.”

The second SSM parameter is amplitude or \(a\), which is calculated as the difference between the highest point of the curve and the curve’s mean. It is interpreted as the distinctiveness or differentiation of a profile: how much it is peaked versus flat. Similar to elevation, when using correlation-based SSM, \(a\ge.15\) is considered “marked” and \(a<.15\) is considered “modest.”

The final SSM parameter is displacement or \(d\), which is calculated as the angle at which the curve reaches its highest point. It is interpreted as the style of the profile. For instance, if \(d=90^\circ\) and we are using a circumplex scale that defines 90 degrees as “domineering,” then the profile’s style is domineering.

By interpreting these three parameters, we can understand a profile much more parsimoniously than by trying to interpret all eight subscales individually. This approach also leverages the circumplex relationship (i.e., dependency) among subscales. It is also possible to transform the amplitude and displacement parameters into estimates of distance from the x-axis and y-axis, which will be shown in the output discussed below.

Conducting SSM for a group’s mean scores

To begin, let’s say that we want to use the SSM to describe the interpersonal problems of the average individual in the entire dataset. Although it is possible to analyze the raw scores contained in jz2017, our results will be more interpretable if we standardize the scores first. We can do this using the standardize() function. The first argument to this function is .data, a data frame frame containing the circumplex scales to be standardized. The second argument is scales and specifies where in .data the circumplex scales are (either in terms of their variable names or their column numbers). The third argument is angles and specifies the angle of each of the circumplex scales included in scales. Note that the scales and angles arguments need to be vectors (hence the c() function) that have the same ordering and length. Finally, the fourth argument is norms, a data frame containing the normative data we will use to standardize the circumplex scales. Here, we will use normative data for the IIP-SC by loading the iipsc data frame.

data("iipsc")
jz2017s <- standardize(
  .data = jz2017,
  scales = c(PA, BC, DE, FG, HI, JK, LM, NO),
  angles = c(90, 135, 180, 225, 270, 315, 360, 45),
  instrument = iipsc,
  sample = 1
)
print(jz2017s)
#> # A tibble: 1,166 x 27
#>    Gender    PA    BC    DE    FG    HI    JK    LM    NO PARPD SCZPD SZTPD
#>    <fct>  <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> <int> <int> <int>
#>  1 Female  1.5   1.5   1.25  1     2     2.5   2.25  2.5      4     3     7
#>  2 Female  0     0.25  0     0.25  1.25  1.75  2.25  2.25     1     0     2
#>  3 Female  0     0     0     0     0     0     0     0        0     1     0
#>  4 Male    2     1.75  1.75  2.5   2     1.75  2     2.5      1     0     0
#>  5 Female  0.25  0.5   0.25  0     0     0     0     0        0     0     0
#>  6 Male    1.5   1.75  2.25  1.75  2     1.25  2.25  2.5      5     5     7
#>  7 Male    2     1.75  1.75  2     1.5   1.25  1.25  1.75     3     6     7
#>  8 Female  0     0.25  0.25  0.25  1     1.25  1.25  1        0     0     1
#>  9 Female  0     0     0     0     0.25  0.25  1     0        0     1     0
#> 10 Male    0.75  2     1.75  2.5   2     1.5   2     2.25     4     5     5
#> # ... with 1,156 more rows, and 15 more variables: ASPD <int>, BORPD <int>,
#> #   HISPD <int>, NARPD <int>, AVPD <int>, DPNPD <int>, OCPD <int>, PA_z <dbl>,
#> #   BC_z <dbl>, DE_z <dbl>, FG_z <dbl>, HI_z <dbl>, JK_z <dbl>, LM_z <dbl>,
#> #   NO_z <dbl>

Now we can use the ssm_analyze() function to perform the SSM analysis. The first three arguments are the same as the first three arguments to standardize(). We can pass the new jz2017s data frame that contains standardized scores as .data and the same vectors to scales and angles since these haven’t changed.

results <- ssm_analyze(
  .data = jz2017s,
  scales = c(PA_z, BC_z, DE_z, FG_z, HI_z, JK_z, LM_z, NO_z),
  angles = c(90, 135, 180, 225, 270, 315, 360, 45)
)

The output of the function has been saved in the results object, which we can examine in detail using the summary() function. This will output the call we made to create the output, as well as some of the default options that we didn’t bother changing (see ?ssm_analyze to learn how to change them) and, most importantly, the estimated SSM parameter values with bootstrapped confidence intervals.

summary(results)
#> Call:
#> ssm_analyze(.data = jz2017s, scales = c(PA_z, BC_z, DE_z, FG_z, 
#>     HI_z, JK_z, LM_z, NO_z), angles = c(90, 135, 180, 225, 270, 
#>     315, 360, 45))
#> 
#> Statistical Basis:    Mean Scores 
#> Bootstrap Resamples:  2000 
#> Confidence Level:     0.95 
#> Listwise Deletion:    TRUE 
#> Scale Displacements:  90 135 180 225 270 315 360 45 
#> 
#> Profile [All]:
#>                Estimate   Lower CI   Upper CI
#> Elevation        -0.225     -0.258     -0.191
#> X-Value           0.131      0.097      0.161
#> Y-Value          -0.015     -0.048      0.020
#> Amplitude         0.131      0.098      0.165
#> Displacement    353.525    339.484      9.564
#> Model Fit         0.710

That was pretty easy! We can now write up these results. However, the circumplex package has some features that can make what we just did even easier. First, because the first three arguments of the ssm_analyze() function are always the same, we can omit their names. Second, because we organized the jz2017s data frame to have the circumplex scale variables adjacent and in order from PA to NO, we can simplify their specification by using the PA:NO shortcut. Finally, because the use of octant scales is so common, the circumplex package comes with a convenience function for outputting their angular displacements: octants(). Note how, even when using these shortcuts, the results are the same except for minor stochastic differences in the confidence intervals due to the randomness inherent to bootstrapping. (To get the exact same results, we could use the set.seed() function to control the random number generator in R.)

results2 <- ssm_analyze(jz2017s, PA_z:NO_z, octants())
summary(results2)
#> Call:
#> ssm_analyze(.data = jz2017s, scales = PA_z:NO_z, angles = octants())
#> 
#> Statistical Basis:    Mean Scores 
#> Bootstrap Resamples:  2000 
#> Confidence Level:     0.95 
#> Listwise Deletion:    TRUE 
#> Scale Displacements:  90 135 180 225 270 315 360 45 
#> 
#> Profile [All]:
#>                Estimate   Lower CI   Upper CI
#> Elevation        -0.225     -0.259     -0.192
#> X-Value           0.131      0.100      0.165
#> Y-Value          -0.015     -0.050      0.019
#> Amplitude         0.131      0.101      0.167
#> Displacement    353.525    339.560      8.994
#> Model Fit         0.710

Visualizing the results with a table and figure

Next, we can produce a table to display our results. With only a single set of parameters, this table is probably overkill, but in future analyses we will see how this function saves a lot of time and effort. To create the table, simply pass the results (or results2) object to the ssm_table() function.

ssm_table(results2)

Next, let’s leverage the fact that we are working within a circumplex space by creating a nice-looking circular plot by mapping the amplitude parameter to the points’ distance from the center of the circle and the displacement parameter to the points’ rotation from due-east (as is conventional). This, again, is as simple as passing the results object to the ssm_plot() function.

ssm_plot(results2)

Conducting SSM for a group’s correlations with an external measure

Next, let’s say that we are interested in analyzing not the mean scores on the circumplex scales but rather their correlations with an external measure. This is sometimes referred to as “projecting” that external measure into the circumplex space. As an example, let’s project the NARPD variable, which captures symptoms of narcissistic personality disorder, into the circumplex space defined by the IIP-SC. Based on theory and previous findings, we can expect this measure to be associated with some general interpersonal distress and a style that is generally high in agency.

To conduct this analysis, we can start with the syntax from the mean-based analysis. All SSM analyses use the ssm_analyze() function and the data, scales, and angles are the same as before. However, we also need to let the function know that we want to analyze correlations with NARPD as opposed to scale means. To do this, we add an additional argument measures. Note that since correlations are already standardized, we don’t need to worry about standardizing the circumplex scales when measures is used.

results3 <- ssm_analyze(jz2017, PA:NO, octants(), measures = NARPD)
summary(results3)
#> Call:
#> ssm_analyze(.data = jz2017, scales = PA:NO, angles = octants(), 
#>     measures = NARPD)
#> 
#> Statistical Basis:    Correlation Scores 
#> Bootstrap Resamples:  2000 
#> Confidence Level:     0.95 
#> Listwise Deletion:    TRUE 
#> Scale Displacements:  90 135 180 225 270 315 360 45 
#> 
#> Profile [NARPD]:
#>                Estimate   Lower CI   Upper CI
#> Elevation         0.202      0.170      0.237
#> X-Value          -0.062     -0.096     -0.028
#> Y-Value           0.179      0.144      0.211
#> Amplitude         0.189      0.155      0.224
#> Displacement    108.967     98.720    118.827
#> Model Fit         0.957

Note that this output looks very similar to the mean-based output except that the statistical basis is now correlation scores instead of mean scores and instead of saying “Profile [All]” it now says “Profile [NARPD]”.

Visualizing the results with a table and figure

We can also create a similar table and figure using the exact same syntax as before. The ssm_table() and ssm_plot() functions are smart enough to know whether the results are mean-based or correlation-based and will work in both cases.

ssm_table(results3)

From the table, we can see that the model fit is good (>.80) and that all three SSM parameters are significantly different from zero, i.e., their confidence intervals do not include zero. Furthermore, the confidence intervals for the elevation and amplitude parameters are greater than or equal to 0.15, which can be interpreted as being “marked.” So, consistent with our hypotheses, NARPD was associated with marked general interpersonal distress (elevation) and was markedly distinctive in its profile (amplitude). The displacement parameter was somewhere between 100 and 120 degrees; to interpret this we would need to either consult the mapping between scales and angles or plot the results.

ssm_plot(results3)

From this figure, it is very easy to see that, consistent with our hypotheses, the displacement for NARPD was associated with high agency and was somewhere between the “domineering” and “vindictive” octants.