## Objective

This vignette demonstrates three major functions in the brokenstick package: brokenstick(), predict() and plot(). We also need dplyr and ggplot2.

require("brokenstick")
require("dplyr")
require("ggplot2")

For more elaborate documentation, see the manual.

## Plot trajectories

The smocc_200 data in the brokenstick package contain the heights of 200 Dutch children measured on 10 visits at ages 0-2 years.

data <- brokenstick::smocc_200
head(data, 3)
## # A tibble: 3 × 7
##      id    age sex       ga    bw   hgt hgt_z
##   <dbl>  <dbl> <chr>  <dbl> <dbl> <dbl> <dbl>
## 1 10001 0      female    40  3960  52   0.575
## 2 10001 0.0821 female    40  3960  55.6 0.888
## 3 10001 0.159  female    40  3960  58.2 0.797

Figure 1 dispays the data from the first 500 rows as a set of growth curves of Dutch children. Curves are steeper during the first few months, so child growth is faster for young infants. Note also there are more cross-overs during the first half year, whereas fewer occur later. This means that the relative positions have been settled by the age of 2 years, or - put differently - that the correlation between time points at those ages is high.

ggplot(data[1:500, ], aes(x = age, y = hgt_z, group = id, color = as.factor(id))) +
geom_line(size = 0.1) +
geom_point(size = 0.7) +
scale_colour_viridis_d(option = "cividis") +
xlab("Age (years)") +
ylab("Length SDS") +
theme_light() +
theme(legend.position = "none")

Figure 2 dispays the same data, but with the vertical axis changed to Standard Deviation Scores (SDS), or $$Z$$-score. The $$Z$$-score is the height corrected for age relative to the Dutch height reference from the Fourth Dutch Growth Study. The $$Z$$-score transformation takes away the major time trend, so all curves are more or less flat. This allows us to see a more detailed assessment of individual growth.

The plots also show how the measurements are clustered around ten ages: birth, 1, 2, 3, 6, 9, 12, 15, 18 and 24 months. While the design was followed rigorously in the study, some variation in timing is inevitable because of weekends, holidays, sickness, and other events. The timing variation poses a problem because we cannot directly compare the measurement between different children (especially for figure 1). Also, we cannot easily construct the “broad” matrix with 10 time point per child.

Of course, we can divide the time axis into ten age groups, and treat all point within the same age group as being measured at the same point. This is probably a good strategy for nicely looking data - as we have here -, but this approach is problematic in data with irregular time intervals, of there are multiple measurement per age group, if the measurement schedules vary by child, or in data combined from studies that employed different designs.

The brokenstick package contains tools to approximate the observed data by a series of connecting straight lines. When these lines closely follow the data, we may replace each trajectory by its values at the breakpoints. The statistical analysis can then be done on the regularised trajectories, which is easier than working with the observed data.

## Fit broken stick model with one line

We fit a trivial broken stick model with just one line on ages between birth and two years, and plot the trajectories of three selected children as follows:

set.seed(123)
fit <- brokenstick(hgt ~ age | id, data, knots = c(0, 2))
ids <- c(10001, 10005, 10022)
plot(fit, group = ids,
xlab = "Age (years)", ylab = "Length (cm)")

The following plot displays the same data, but in standardised units so as to increase the analytic resolution:

fit0 <- brokenstick(hgt_z ~ age | id, data, knots = c(0, 2))
plot(fit0, group = ids,
xlab = "Age (years)", ylab = "Length (SDS)")

Note that both approximations describe the individual trend in the data, but do not address any systematic deviations from the trend.

## Fit broken stick model with two lines

The broken stick model describes a trajectory by a series of connected straight lines. We first calculate a model with two connected lines. The first line starts at birth and end at the age of exactly 1 years. The second line spans the period between 1 to 2 years. In addition, the lines must connect at the age of 1 year. We estimate and plot the model as follows:

fit2 <- brokenstick(hgt_z ~ age | id, data = data, knots = c(0, 1, 2))
plot(fit2, group = ids, xlab = "Age (years)", ylab = "Length (SDS)")

The plot shows that the two-line model is still fairly crude. The fit2 object holds the parameter estimates of the model:

summary(fit2)
## Class        brokenstick (kr)
## Variables    hgt_z (outcome), age (predictor), id (group)
## Data         1942 (n), 36 (nmis), 200 (groups)
## Parameters   16 (total), 4 (fixed), 4 (variance), 6 (covariance), 2 (error)
## Knots        0 1 2
## Means        -0.0394  0.0278  0.0646
## Residuals    0.113 0.146 0.168 0.195 0.383 (min, P25, P50, P75, max)
## Mean resid   0.179
## R-squared    0.867
##
## Variance-covariance matrix
##       age_0 age_1 age_2
## age_0 1.218
## age_1 0.481 0.823
## age_2 0.485 0.782 0.884

The console output lists the knots of the model, including the left and right boundary knots at 0 and 2.6776. The row of means correspond to the fixed effect estimates of the linear mixed model. We may interpret these as the global means. Next, the output lists the variance-covariance matrix of the random effects. The model contains 16 parameters in total: four fixed effects (means), four random effects (diagonal elements), 6 covariance (off-diagonal elements) and 2 error variances (one for the residual error variance, one for the variability of the error per cluster). These parameters are enough to reconstruct the broken stick model, and to apply it to new data.

## Extend to nine lines

We refine the model in the first two years by adding a knot for each age at which a visit was scheduled. This model can be run as

knots <- round(c(0, 1, 2, 3, 6, 9, 12, 15, 18, 24)/12, 4)
fit9 <- brokenstick(hgt_z ~ age | id, data = data, knots = knots)

This optimization problem is more difficult, so it takes slightly longer to run. The results are

summary(fit9)
## Class        brokenstick (kr)
## Variables    hgt_z (outcome), age (predictor), id (group)
## Data         1942 (n), 36 (nmis), 200 (groups)
## Parameters   79 (total), 11 (fixed), 11 (variance), 55 (covariance), 2 (error)
## Knots        0.0000 0.0833 0.1667 0.2500 0.5000 0.7500 1.0000 1.2500 1.5000 2.0000
## Means        -0.18618  0.02163  0.00262  0.06602  0.06346 -0.01955  0.00781  0.03233 -0.02665  0.11470
## Residuals    0.0176 0.0268 0.0329 0.0494 4.5190 (min, P25, P50, P75, max)
## Mean resid   0.0752
## R-squared    0.983
##
## Variance-covariance matrix
##            age_0 age_0.0833 age_0.1667 age_0.25 age_0.5 age_0.75 age_1 age_1.25
## age_0      1.740
## age_0.0833 1.118      1.337
## age_0.1667 1.077      1.095      1.246
## age_0.25   0.900      1.005      0.999    1.052
## age_0.5    0.627      0.821      0.822    0.815   0.943
## age_0.75   0.597      0.696      0.693    0.718   0.867    0.938
## age_1      0.495      0.611      0.579    0.606   0.786    0.819 0.869
## age_1.25   0.432      0.614      0.575    0.601   0.757    0.807 0.834    0.953
## age_1.5    0.455      0.614      0.624    0.604   0.808    0.824 0.836    0.914
## age_2      0.361      0.483      0.517    0.539   0.720    0.753 0.783    0.883
##            age_1.5 age_2
## age_0
## age_0.0833
## age_0.1667
## age_0.25
## age_0.5
## age_0.75
## age_1
## age_1.25
## age_1.5      1.039
## age_2        0.919  1.06

There nine-line model summarises the data by 79 parameters. This model fits substantially better. The model residuals are substantially smaller (0.075 instead of 0.179) and the proportions of explained variance is much higher (0.983 instead of 0.867).

The figure shows that the nine-line broken stick model fits the observed data very well.

## Obtain predicted values

The predict() function allows us to obtain various types of predictions from the broken stick model. The simplest call

p1 <- predict(fit2)
head(p1)
##   .pred
## 1 0.720
## 2 0.660
## 3 0.604
## 4 0.534
## 5 0.352
## 6 0.171
identical(nrow(data), nrow(p1))
## [1] TRUE

produces a tibble with the one column called .pred for each row in data. We can bind column .pred to data for further processing.

Sometimes, we also want the prediction at the knot values, for example, to create graphs that contain observed and modelled trajectories. We obtain predictions at the knots by the special x = "knots" argument, e.g.

p2 <- predict(fit2, x = "knots", include_data = FALSE)
head(p2)
##   .source    id age  sex ga bw hgt hgt_z    .pred
## 1   added 10001   0 <NA> NA NA  NA    NA  0.71989
## 2   added 10001   1 <NA> NA NA  NA    NA -0.00939
## 3   added 10001   2 <NA> NA NA  NA    NA  0.05785
## 4   added 10002   0 <NA> NA NA  NA    NA -0.23264
## 5   added 10002   1 <NA> NA NA  NA    NA -0.37772
## 6   added 10002   2 <NA> NA NA  NA    NA -0.50233
nrow(p2)
## [1] 600

We use the include_data = FALSE argument to remove predictions for the observed data. The output is more verbose and includes the grid of knots for each child (id, age). The column .source is equal to added as all rows are non-observed data.

Note there are also knots at ages 0.00 and 2.68 years. These are boundary knots, and added by the brokenstick() function. The boundary knots effectively filter the observations that enter the calculations. By default, the boundary knots span the age range in the data. For technical reasons, the broken stick model also defines and estimates parameters for these knots, but these may in general be ignored, especially when the data near the boundary knots are sparse.

If we wish to obtain estimates at both the knots and the observed data use:

p3 <- predict(fit2, x = "knots", hide = "none")
table(p3\$.source)
##
##   800  1942

This return 1942 rows for the data and 800 rows for the knots.

## Explained variance

The proportion of the variance of the outcome explained by the two-line model is

get_r2(fit2)
## [1] 0.867

For the second model we get

get_r2(fit9)
## [1] 0.983

so the nine-line broken stick model explains about 98 percent of the variance of the height SDS.

## Subject level analysis

Suppose we are interest in knowing the effect of sex, gestational age and birth weight on the height SDS at the age of 2 years. This is an analysis at the subject level. Let us first extract the subject-level data with variables that vary over subjects only.

subj <- data %>%
select(id, sex, ga, bw) %>%
group_by(id) %>%
slice(1)
head(subj, 3)
## # A tibble: 3 × 4
## # Groups:   id [3]
##      id sex       ga    bw
##   <dbl> <chr>  <dbl> <dbl>
## 1 10001 female    40  3960
## 2 10002 male      38  3210
## 3 10003 female    40  4170

We also need the outcome variable. We take it from the broken stick estimates from the nine line solution and append it to the subject level data.

bs <- predict(fit9, x = "knots", shape = "wide", include_data = FALSE)
data <- bind_cols(subj, select(bs, -id))
head(data, 3)
## # A tibble: 3 × 14
## # Groups:   id [3]
##      id sex       ga    bw    0 0.0833 0.1667 0.25  0.5 0.75    1
##   <dbl> <chr>  <dbl> <dbl>  <dbl>    <dbl>    <dbl>  <dbl>  <dbl>  <dbl>  <dbl>
## 1 10001 female    40  3960  0.569    0.865    0.735  0.657  0.232 -0.172  0.118
## 2 10002 male      38  3210 -0.184   -0.329   -0.252 -0.319 -0.218 -0.281 -0.344
## 3 10003 female    40  4170  1.17     2.11     1.97   2.04   2.03   1.81   1.32
## # … with 3 more variables: 1.25 <dbl>, 1.5 <dbl>, 2 <dbl>

The names of the columns in bs correspond to the knot values.

The effect of the subject’s sex, gestational age and birth weight on the height SDS at the age of 2 years (here denoted by the variable named 2) can be estimated as

fit1_lm <- lm(2 ~ sex + ga + I(bw / 1000), data = data)
summary(fit1_lm)
##
## Call:
## lm(formula = 2 ~ sex + ga + I(bw/1000), data = data)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -2.8965 -0.4525  0.0817  0.6424  2.0369
##
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)
## (Intercept)   1.0361     1.4705    0.70  0.48187
## sexmale       0.0109     0.1285    0.08  0.93249
## ga           -0.0700     0.0433   -1.62  0.10752
## I(bw/1000)    0.5343     0.1420    3.76  0.00022 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.879 on 196 degrees of freedom
## Multiple R-squared:  0.0757, Adjusted R-squared:  0.0616
## F-statistic: 5.35 on 3 and 196 DF,  p-value: 0.00145

Note that the analysis shows there is a substantial effect of birth weight. Of course, it might be that birth weight is directly related to height at the age of 2 years. Alternatively, the relation could be mediated by birth length. The following model adds birth length (the variable named 0) to the model:

fit2_lm <- lm(2 ~ sex + ga + I(bw / 1000) + 0, data = data)
summary(fit2_lm)
##
## Call:
## lm(formula = 2 ~ sex + ga + I(bw/1000) + 0, data = data)
##
## Residuals:
##     Min      1Q  Median      3Q     Max
## -2.9375 -0.5326  0.0426  0.6292  1.8792
##
## Coefficients:
##              Estimate Std. Error t value Pr(>|t|)
## (Intercept)  3.226719   1.662005    1.94   0.0536 .
## sexmale     -0.000703   0.126542   -0.01   0.9956
## ga          -0.089472   0.043247   -2.07   0.0399 *
## I(bw/1000)   0.143924   0.201777    0.71   0.4765
## 0          0.245522   0.091492    2.68   0.0079 **
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## Residual standard error: 0.865 on 195 degrees of freedom
## Multiple R-squared:  0.109,  Adjusted R-squared:  0.0904
## F-statistic: 5.94 on 4 and 195 DF,  p-value: 0.000156

The effect of birth length on length at age 2 is very strong. There is no separate effect of birth weight anymore, so this analysis suggests that the relation between birth weight and length at age 2 can be explained by their mutual associations to birth length.

## Conclusion

This vignette illustrated the use of the brokenstick(), plot() and predict() functions. Other vignettes highlight various other capabilities of the package.