opa

Data

For this example we will use childhood growth data reported by Potthoff & Roy (1964) consisting of measures of the distance between the pituitary and the pteryo-maxillary fissure. Distances were recorded in millimeters when each child was 8, 10, 12 and 14 years old. This is same data set available as Orthodont from the nlme package.

str(pituitary)
#> 'data.frame':    108 obs. of  4 variables:
#>  $ distance  : num  26 25 29 31 21.5 22.5 23 26.5 23 22.5 ...
#>  $ age       : int  8 10 12 14 8 10 12 14 8 10 ...
#>  $ individual: Factor w/ 27 levels "1","2","3","4",..: 1 1 1 1 2 2 2 2 3 3 ...
#>  $ sex       : Factor w/ 2 levels "F","M": 2 2 2 2 2 2 2 2 2 2 ...

xyplot(distance ~ age | individual, pituitary, type = c("g", "p", "l"),
       groups = sex, cex = 1.2, pch = 21,
       fill = c("#EFC00070", "#0073C270"), col = c("#EFC000", "#0073C2"),
       scales = list(x = list(at = c(8, 10, 12, 14))),
       xlab = "Age (years)", ylab = "Distance (mm)",
       main = "Pituitary-Pterygomaxillary Fissure Distance",
       key = list(space = "bottom", columns = 2,
                  text = list(c("Female", "Male")),
                  lines = list(col = c("#EFC000", "#0073C2")),
                  points = list(pch = 21, fill = c("#EFC00070", "#0073C270"),
                                col = c("#EFC000", "#0073C2"))))

dat_wide <- reshape(data = pituitary,
                    direction = "wide",
                    idvar = c("individual", "sex"),
                    timevar = "age",
                    v.names = "distance")

head(dat_wide)
#>    individual sex distance.8 distance.10 distance.12 distance.14
#> 1           1   M       26.0        25.0        29.0        31.0
#> 5           2   M       21.5        22.5        23.0        26.5
#> 9           3   M       23.0        22.5        24.0        27.5
#> 13          4   M       25.5        27.5        26.5        27.0
#> 17          5   M       20.0        23.5        22.5        26.0
#> 21          6   M       24.5        25.5        27.0        28.5

Specifying a hypothesis

For this analysis we will assume a hypothesis of monotonic increase in distance with increasing age. This monotonic increase hypothesis can be encoded using the hypothesis() function.

h1 <- hypothesis(c(1, 2, 3, 4), type = "pairwise")

Printing the hypothesis object shows that it consists of sub-hypotheses about six ordinal relations. In this case it is hypothesized that each of the six ordinal relations are increases, coded as 1.

print(h1)
#> ********** Ordinal Hypothesis **********
#> Hypothesis type: pairwise 
#> Raw hypothesis:
#>  1 2 3 4 
#> Ordinal relations:
#>  1 1 1 1 1 1 
#> N conditions: 4 
#> N hypothesised ordinal relations: 6 
#> N hypothesised increases: 6 
#> N hypothesised decreases: 0 
#> N hypothesised equalities: 0

The hypothesis can also be visualized with the plot() function. Note that the y-axis is unitless; the vertical distance between points has no meaning. The information represented by the y-axis is relative: bigger, smaller or equal.

plot(h1)

Fitting an ordinal pattern analysis model

How well a hypothesis accounts for observed data can be quantified at the individual and group levels using the opa() function. The first required argument to opa() is a dataframe consisting of only the response variable columns. The second required argument is the hypothesis.

m1 <- opa(dat_wide[,3:6], h1)

The results can be summarized using the summary() function.

summary(m1)
#> Ordinal Pattern Analysis of 4 observations for 27 individuals in 1 group 
#> 
#> Between subjects results:
#>          PCC   cval
#> pooled 88.27 <0.001
#> 
#> Within subjects results:
#>       PCC cval
#> 1   83.33 0.17
#> 2  100.00 0.04
#> 3   83.33 0.17
#> 4   66.67 0.38
#> 5   83.33 0.17
#> 6  100.00 0.04
#> 7   83.33 0.08
#> 8   83.33 0.19
#> 9   66.67 0.41
#> 10 100.00 0.04
#> 11  83.33 0.09
#> 12 100.00 0.04
#> 13 100.00 0.05
#> 14  83.33 0.08
#> 15 100.00 0.04
#> 16  83.33 0.16
#> 17  83.33 0.17
#> 18 100.00 0.04
#> 19 100.00 0.04
#> 20 100.00 0.04
#> 21  83.33 0.19
#> 22  83.33 0.09
#> 23 100.00 0.05
#> 24  83.33 0.07
#> 25  83.33 0.15
#> 26  83.33 0.09
#> 27  83.33 0.09
#> 
#> PCCs were calculated for pairwise ordinal relationships using a difference threshold of 0.
#> Chance-values were calculated from 1000 random orderings.

The individual-level results can also be visualized using plot().

plot(m1)

It is also possible to determine how well the hypothesis accounts for the data at the group level for each of the sub-hypotheses using the compare_conditions() function.

compare_conditions(m1)
#> Pairwise PCCs:
#>        1      2      3 4
#> 1      -      -      - -
#> 2 66.667      -      - -
#> 3    100 77.778      - -
#> 4    100 96.296 88.889 -
#> 
#> Pairwise chance values:
#>        1      2      3 4
#> 1      -      -      - -
#> 2  0.014      -      - -
#> 3 <0.001 <0.001      - -
#> 4 <0.001 <0.001 <0.001 -

These results indicate that the hypothesis accounts least well for the relationship between the first two measurement times.

Comparing hypotheses

The output of compare_conditions() indicates that it may be informative to consider a second, non-monotonic hypothesis:

h2 <- hypothesis(c(2, 1, 3, 4))

h2 contains the sub-hypothesis that the second measurement, at age 10, may be shorter than the first measurement taken at age 8.

plot(h2)

The hypothesized decrease between measurements at ages 8 and 10 is encoded as -1.

print(h2)
#> ********** Ordinal Hypothesis **********
#> Hypothesis type: pairwise 
#> Raw hypothesis:
#>  2 1 3 4 
#> Ordinal relations:
#>  -1 1 1 1 1 1 
#> N conditions: 4 
#> N hypothesised ordinal relations: 6 
#> N hypothesised increases: 5 
#> N hypothesised decreases: 1 
#> N hypothesised equalities: 0

To assess the adequacy of h2 we fit a second opa() model, this time passing h2 as the second argument.

m2 <- opa(dat_wide[,3:6], h2)

The compare_hypotheses() function can then be used to compare how well two hypothese account for the observed data.

(comp_h1_h2 <- compare_hypotheses(m1, m2))
#> ********* Hypothesis Comparison **********
#> H1: 1 2 3 4 
#> H2: 2 1 3 4 
#> H1 PCC: 88.2716 
#> H2 PCC: 80.8642 
#> PCC difference: 7.407407 
#> cval: 0.275 
#> Comparison type: two-tailed

These results indicate that the monotonic increase hypothesis encoded by h1 better acounts for the data than h2. However, the difference between these hypotheses is not large. The computed chance value indicates that a difference at least as large could be produced by chance, through random permutations of the data, about one quarter of the time. The calculation of the group-level chance-value can be visualized by plotting the object returned by compare_hypotheses().

plot(comp_h1_h2)

Comparing groups

So far the analyses have not considered any possible differences between males and females in terms of how well the hypotheses account for the data. It is possible that a given hypothesis account better for males than for females, or vice-versa. A model which accounts for groups within the data can be fitted by passing a factor vector or dataframe column to the group argument.

m3 <- opa(dat_wide[,3:6], h1, group = dat_wide$sex)

The plot() function can be used to get a first sense of how PCCs and chance-values differ between groups.

plot(m3)

There is not a clearly visible pattern that appears to distinguish males from females. The compare_groups() function may be used to more precisely quantify the difference in hypothesis performance between the groups within the data.

(comp_m_f <- compare_groups(m3, "M", "F"))
#> ********* Group Comparison **********
#> Group 1: M 
#> Group 2: F 
#> Group 1 PCC: 87.5 
#> Group 2 PCC: 89.39394 
#> PCC difference: 1.893939 
#> cval: 0.849 
#> Comparison type: two-tailed

In this case, compare_groups() indicates that the difference in how well the hypothesis accounts for males and females growth is very small. The chance value shows that a difference at least as great could be produced by chance around 80% of the time. As with the hypothesis comparison, the computation of the chance value for the group comparison can be visualized by plotting the object returned by compare_groups().

plot(comp_m_f)

Difference thresholds

Each of the above models has treated any numerical difference in the distance data as a true difference. However, we may wish to treat values that differ by some small amount as equivalent. In this way we may define a threshold of practical or clinical significance. For example, we may decide to consider only differences of at least 1 mm. This can be achieved by passing a threshold value of 1 using the diff_threshold argument.

m4 <- opa(dat_wide[,3:6], h1, group = dat_wide$sex, diff_threshold = 1)

Setting the difference threshold to 1 mm results in a greater difference in hypothesis performance between males and females.

group_results(m4)
#>     PCC   cval
#> F 54.55 <0.001
#> M 71.88 <0.001

However, compare_groups() reveals that even this larger difference could occur quite frequently by chance.

compare_groups(m4, "M", "F")
#> ********* Group Comparison **********
#> Group 1: M 
#> Group 2: F 
#> Group 1 PCC: 71.875 
#> Group 2 PCC: 54.54545 
#> PCC difference: 17.32955 
#> cval: 0.21 
#> Comparison type: two-tailed