Using surveil for public health research
This vignette demonstrates basic usage of surveil for public health research. The package is designed for routine time trend analysis, namely for time trends in disease incidence rates or mortality rates. Models were built using the Stan modeling language for Bayesian inference with Markov chain Monte Carlo (MCMC), but users only need to be familiar with the R language.
The package also contains special methods for age-standardization,
printing and plotting model results, and for measuring and visualizing
health inequalities. For age-standardization see
vignette("age-standardization"). For discussion and
demonstration analysis see Donegan et al. (2022).
Getting started
To use the models provided by surveil, the surveillance data minimally must contain case counts, population at risk estimates, and a discrete time period variable. The data may also include one or more grouping variables, such as race-ethnicity. Time periods should consist of equally spaced intervals.
This vignette analyzes colorectal cancer incidence data by race-ethnicity, year, and Texas MSA for ages 50-79 (data obtained from CDC Wonder). The race-ethnicity grouping includes (non-Hispanic) black, (non-Hispanic) white, and Hispanic, and the MSAs include those centered on the cities of Austin, Dallas, Houston, and San Antonio.
head(msa) |>
kable(booktabs = TRUE,
caption = "Glimpse of colorectal cancer incidence data (CDC Wonder)") | Year | Race | MSA | Count | Population |
|---|---|---|---|---|
| 1999 | Black or African American | Austin-Round Rock, TX | 28 | 14421 |
| 2000 | Black or African American | Austin-Round Rock, TX | 16 | 15215 |
| 2001 | Black or African American | Austin-Round Rock, TX | 22 | 16000 |
| 2002 | Black or African American | Austin-Round Rock, TX | 24 | 16694 |
| 2003 | Black or African American | Austin-Round Rock, TX | 34 | 17513 |
| 2004 | Black or African American | Austin-Round Rock, TX | 26 | 18429 |
The primary function in surveil is stan_rw,
which fits random walk models to surveillance data. The function is
expects the user to provide a data.frame with specific
column names. There must be one column named Count
containing case counts, and another column named
Population, containing the sizes of the populations at
risk. The user must provide the name of the column containing the time
period (the default is time = Year, to match CDC Wonder
data). Optionally, one can provide a grouping factor. For the MSA data
printed above, the grouping column is Race and the time column is
Year.
Preparing the data
We will demonstrate using aggregated CRC cases across Texas’s top
four MSAs. The msa data from CDC Wonder already has the
necessary format (column names and contents), but these data are
dis-aggregated by MSA. So for this analysis, we first group the data by
year and race, and then combine cases across MSAs.
The following code chunk aggregates the data by year and race-ethnicity:
The following code provides a glimpse of the aggregated data:
head(msa2) |>
kable(booktabs = TRUE,
caption = "Glimpse of aggregated Texas metropolitan CRC cases, by race and year")| Year | Race | Count | Population |
|---|---|---|---|
| 1999 | Black or African American | 471 | 270430 |
| 2000 | Black or African American | 455 | 283280 |
| 2001 | Black or African American | 505 | 298287 |
| 2002 | Black or African American | 539 | 313133 |
| 2003 | Black or African American | 546 | 329481 |
| 2004 | Black or African American | 602 | 346886 |
Model specification
The basics
The base surveil model is specified as follows. The Poisson model is used as the likelihood: the probability of observing a given number of cases, \(y_t\), conditional on a given level of risk, \(e^{\phi_t}\), and known population at risk, \(p_t\), is: \[y_t \sim \text{Pois}(p_t \cdot e^{\phi_t})\] where \(t\) indexes the time period.
Next, we need a model for the log-rates, \({\phi_t}\). The first-difference prior states that our expectation for the log-rate at any time is its previous value, and we assign a normal probability distribution to deviations from the previous value (Clayton 1996). This is also known as the random-walk prior: \[\phi_t \sim \text{Gau}(\phi_{t-1}, \tau^2)\] This places higher probability on a smooth trend through time, specifically implying that underlying disease risk tends to have less variation than crude incidence.
The log-risk for time \(t=1\) has no previous value to anchor its expectation; thus, we assign a prior probability distribution directly to \(\phi_1\). For this prior, surveil uses a normal distribution. The scale parameter, \(\tau\), also requires a prior distribution, and again surveil uses a normal model which is diffuse relative to the log incidence rates.
Binomial model
In addition to the Poisson model, the binomial model is also available: \[y_t \sim \text{Binom}(p_t \cdot g^{-1}(\phi_t))\] where \(g\) is the logit function and \(g^{-1}(x) = \frac{exp(x)}{1 + exp(x)}\) (the inverse-logit function). If the binomial model is used the rest of the model remains the same as stated above. The Poisson model is typically preferred for rare events (such as rates below .01), otherwise the binomial model is usually more appropriate. The remainder of this vignette will proceed using the Poisson model only.
Fitting the model
The time series model is fit by passing surveillance data to the
stan_rw function. Here, Year and
Race indicate the appropriate time and grouping columns in
the msa2 data frame.
fit <- stan_rw(msa2, time = Year, group = Race, iter = 1e3)
#> Distribution: normal
#> Distribution: normal
#> [1] "Setting normal prior(s) for eta_1: "
#> location scale
#> -6 5
#> [1] "\nSetting half-normal prior for sigma: "
#> location scale
#> 0 1
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 1).
#> Chain 1:
#> Chain 1: Gradient evaluation took 1.6e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.16 seconds.
#> Chain 1: Adjust your expectations accordingly!
#> Chain 1:
#> Chain 1:
#> Chain 1: Iteration: 1 / 1000 [ 0%] (Warmup)
#> Chain 1: Iteration: 501 / 1000 [ 50%] (Sampling)
#> Chain 1: Iteration: 1000 / 1000 [100%] (Sampling)
#> Chain 1:
#> Chain 1: Elapsed Time: 0.266 seconds (Warm-up)
#> Chain 1: 0.149 seconds (Sampling)
#> Chain 1: 0.415 seconds (Total)
#> Chain 1:
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 2).
#> Chain 2:
#> Chain 2: Gradient evaluation took 1.3e-05 seconds
#> Chain 2: 1000 transitions using 10 leapfrog steps per transition would take 0.13 seconds.
#> Chain 2: Adjust your expectations accordingly!
#> Chain 2:
#> Chain 2:
#> Chain 2: Iteration: 1 / 1000 [ 0%] (Warmup)
#> Chain 2: Iteration: 501 / 1000 [ 50%] (Sampling)
#> Chain 2: Iteration: 1000 / 1000 [100%] (Sampling)
#> Chain 2:
#> Chain 2: Elapsed Time: 0.26 seconds (Warm-up)
#> Chain 2: 0.147 seconds (Sampling)
#> Chain 2: 0.407 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1.2e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 3: Adjust your expectations accordingly!
#> Chain 3:
#> Chain 3:
#> Chain 3: Iteration: 1 / 1000 [ 0%] (Warmup)
#> Chain 3: Iteration: 501 / 1000 [ 50%] (Sampling)
#> Chain 3: Iteration: 1000 / 1000 [100%] (Sampling)
#> Chain 3:
#> Chain 3: Elapsed Time: 0.199 seconds (Warm-up)
#> Chain 3: 0.149 seconds (Sampling)
#> Chain 3: 0.348 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1.2e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.12 seconds.
#> Chain 4: Adjust your expectations accordingly!
#> Chain 4:
#> Chain 4:
#> Chain 4: Iteration: 1 / 1000 [ 0%] (Warmup)
#> Chain 4: Iteration: 501 / 1000 [ 50%] (Sampling)
#> Chain 4: Iteration: 1000 / 1000 [100%] (Sampling)
#> Chain 4:
#> Chain 4: Elapsed Time: 0.258 seconds (Warm-up)
#> Chain 4: 0.151 seconds (Sampling)
#> Chain 4: 0.409 seconds (Total)
#> Chain 4:The iter = 1e3 line controls how long the MCMC sampling
continues for (in this case, 1,000 samples: 500 warmup, then 500 more
for inference). The default is 3,000, which is more than sufficient for
this example model. By default, there are four independent MCMC chains
each with 500 post-warmup samples (for a total of 2,000 MCMC samples
used for the estimates).
To speed things up, we could take advantage of parallel processing
using the cores argument (e.g., add cores = 4)
to run on 4 cores simultaneously. You can suppress the messages seen
above by adding refresh = 0.
Printing results
The print method will print the estimates with 95%
credible intervals to the console; adding scale = 100e3
will display rates per 100,000:
print(fit, scale = 100e3)
#> Summary of surveil model results
#> Time periods: 19
#> Grouping variable: Race
#> Correlation matrix: FALSE
#> time Race mean lwr_2.5 upr_97.5
#> 1 1999 Black or African American 170.26634 158.90838 182.31787
#> 2 2000 Black or African American 166.47549 156.09260 176.72263
#> 3 2001 Black or African American 168.30655 158.27298 178.61337
#> 4 2002 Black or African American 169.22421 159.71180 179.63907
#> 5 2003 Black or African American 167.12537 157.79571 176.95320
#> 6 2004 Black or African American 166.82794 157.24391 177.63329
#> 7 2005 Black or African American 159.04921 150.38701 168.47197
#> 8 2006 Black or African American 154.70208 145.90129 162.95879
#> 9 2007 Black or African American 152.62830 144.52354 161.54779
#> 10 2008 Black or African American 149.73023 141.82796 158.60942
#> 11 2009 Black or African American 143.55170 136.12111 151.61151
#> 12 2010 Black or African American 138.91939 131.78328 146.79089
#> 13 2011 Black or African American 130.91745 123.74220 138.37027
#> 14 2012 Black or African American 125.24261 118.57274 132.17447
#> 15 2013 Black or African American 124.82414 117.61514 131.61197
#> 16 2014 Black or African American 126.28177 119.86982 133.20310
#> 17 2015 Black or African American 122.40991 115.99816 128.73548
#> 18 2016 Black or African American 121.94784 115.40496 128.26670
#> 19 2017 Black or African American 122.41853 115.31972 129.50850
#> 20 1999 Hispanic 101.54287 94.47712 108.45484
#> 21 2000 Hispanic 104.12376 98.24256 110.63981
#> 22 2001 Hispanic 102.44859 96.94005 108.32675
#> 23 2002 Hispanic 101.43536 96.20656 106.86244
#> 24 2003 Hispanic 100.50792 95.48950 105.95109
#> 25 2004 Hispanic 100.89734 96.08819 106.63738
#> 26 2005 Hispanic 98.57508 93.73771 103.69278
#> 27 2006 Hispanic 96.85032 91.98685 102.26873
#> 28 2007 Hispanic 94.17633 89.82861 98.72277
#> 29 2008 Hispanic 90.55404 85.58075 94.96131
#> 30 2009 Hispanic 88.85016 84.14400 92.95075
#> 31 2010 Hispanic 88.06195 84.01497 92.26475
#> 32 2011 Hispanic 87.47620 83.24625 91.50249
#> 33 2012 Hispanic 87.83780 84.03505 91.66519
#> 34 2013 Hispanic 87.28182 83.39602 91.36709
#> 35 2014 Hispanic 85.83935 82.02392 89.64141
#> 36 2015 Hispanic 85.85480 82.11206 89.71743
#> 37 2016 Hispanic 84.85297 80.84632 88.55269
#> 38 2017 Hispanic 85.88204 81.71014 90.28463
#> 39 1999 White 135.19672 129.94389 140.56151
#> 40 2000 White 136.50536 131.86437 140.89253
#> 41 2001 White 134.15402 129.62896 138.58170
#> 42 2002 White 130.18419 125.54362 134.75903
#> 43 2003 White 127.49689 122.99644 132.04803
#> 44 2004 White 120.00622 115.91441 124.27156
#> 45 2005 White 115.11728 111.14432 119.27711
#> 46 2006 White 109.69241 105.95299 113.57358
#> 47 2007 White 108.19815 104.51524 112.17423
#> 48 2008 White 103.47740 99.99629 107.24859
#> 49 2009 White 98.91231 95.19104 102.36613
#> 50 2010 White 96.24293 93.04998 99.56098
#> 51 2011 White 93.73885 90.41600 97.08338
#> 52 2012 White 91.11867 87.82641 94.37714
#> 53 2013 White 90.15754 87.07552 93.28108
#> 54 2014 White 91.48918 88.43350 94.73186
#> 55 2015 White 93.10693 90.01396 96.36481
#> 56 2016 White 89.03941 85.96802 91.88921
#> 57 2017 White 92.31623 88.97464 95.85858This information is also stored in a data frame,
fit$summary:
head(fit$summary)
#> time mean lwr_2.5 upr_97.5 Race Year Count
#> 1 1999 0.001702663 0.001589084 0.001823179 Black or African American 1999 471
#> 2 2000 0.001664755 0.001560926 0.001767226 Black or African American 2000 455
#> 3 2001 0.001683065 0.001582730 0.001786134 Black or African American 2001 505
#> 4 2002 0.001692242 0.001597118 0.001796391 Black or African American 2002 539
#> 5 2003 0.001671254 0.001577957 0.001769532 Black or African American 2003 546
#> 6 2004 0.001668279 0.001572439 0.001776333 Black or African American 2004 602
#> Population Crude
#> 1 270430 0.001741671
#> 2 283280 0.001606185
#> 3 298287 0.001693000
#> 4 313133 0.001721313
#> 5 329481 0.001657152
#> 6 346886 0.001735440The fit$summary object can be used to create custom
plots and tables.
Visualizing results
If we call plot on a fitted surveil model, we
get a ggplot object depicting risk estimates with 95%
credible intervals:
The crude incidence rates (observed values) are also plotted here as
points.
The plot method has a number of optional arguments that
control its appearance. For example, the base_size argument
controls the size of labels. The size of the points for the crude rates
can be adjusted using size, and size = 0
removes them altogether. We can also use ggplot to add
custom modifications:
fig <- plot(fit, scale = 100e3, base_size = 11, size = 0)
#> Plotted rates are per 100,000
fig +
theme(legend.position = "right") +
labs(title = "CRC incidence per 100,000",
subtitle = "Texas MSAs, 50-79 y.o.")
The plot method has a style argument that controls how
uncertainty is represented. The default, style = "mean_qi",
shows the mean of the posterior distribution as the estimate and adds
shading to depict the 95% credible interval (as above). The alternative,
style = "lines", plots MCMC samples from the joint
probability distribution for the estimates:
By default, M = 250 samples are plotted. The
style option is available for all of the surveil
plot methods. This style is sometimes helpful for visualizing multiple
groups when their credible intervals overlap.
Percent change
The apc method calculates percent change by period and
cumulatively over time:
The object returned by apc contains two data frames. The
first contains estimates of percent change from the previous period:
head(fit_pc$apc)
#> time group apc lwr upr
#> 1 1999 Black or African American 0.0000000 0.000000 0.000000
#> 2 2000 Black or African American -2.1511874 -9.231349 4.643193
#> 3 2001 Black or African American 1.1626977 -5.432589 8.335396
#> 4 2002 Black or African American 0.6051971 -5.773809 7.979825
#> 5 2003 Black or African American -1.1792908 -7.824318 5.346592
#> 6 2004 Black or African American -0.1281814 -6.302749 6.876742Those estimates typically have high uncertainty.
The second data frame contains estimates of cumulative percent change (since the first observed period):
head(fit_pc$cpc)
#> time group cpc lwr upr
#> 1 1999 Black or African American 0.0000000 0.000000 0.000000
#> 2 2000 Black or African American -2.1511874 -9.231349 4.643193
#> 3 2001 Black or African American -1.0457601 -9.545905 7.405015
#> 4 2002 Black or African American -0.4962134 -8.726262 8.467526
#> 5 2003 Black or African American -1.7258015 -9.967735 7.281973
#> 6 2004 Black or African American -1.9030371 -10.048494 7.202011Each value in the cpc column is an estimate of the
difference in incidence rates between that year and the first year (in
this case, 1999) expressed as a percent of the first year’s rate. The
lwr and upr columns are the lower and upper
bounds of the 95% credible intervals for the estimates.
This information can also be plotted:
If desired, the average annual percent change from the first period can be calculated by dividing the cumulative percent change (CPC) by the appropriate number of periods. For example, the CPC from 1999 to 2017 for whites is about -31 for an average annual percent change of about \(-31/18 = -1.72\). The credible intervals for the average annual percent change can also be obtained from the CPC table using the same method. For this example, the correct denominator is \(2017-1999=18\) (generally: the last year minus the first year).
MCMC diagnostics
If you do not see any warnings printed to the R console at the end of
the model fitting process then you can simply move forward with the
analysis. If there is a warning, it may say that the effective sample
size is low or that the R-hat values are large. For a crash course on
MCMC analysis with surveil, including MCMC diagnostics, see the
vignette on the topic vignette("surveil-mcmc").
A quick and dirty summary is that you want to watch two key
diagnostics, which are effective MCMC sample size (ESS) and R-hat. For
all your parameters of interest, ESS should be at least 400 or so. If
you want to increase the ESS, use the iter argument to draw
more samples. The R-hat values should all be pretty near to one, such as
within the range \(1 \pm .02\). For the
simple models we are using here, large R-hat values can often be fixed
just by drawing more samples.
You can find these diagnostics by printing results
print(fit$samples) and seeing the columns
n_eff (for ESS) and Rhat.
The MCMC sampling is generally fast and without trouble when the
numbers are not too small (as in this example model). When the incidence
rates are based on small numbers (like counts less than 20), the
sampling often proceeds more slowly and a higher iter value
may be needed.
Multiple time series
In most applications, the base model specification described above will be entirely sufficient. However, surveil provides an option for users to add a correlation structure to the model when multiple groups are modeled together. For correlated trends, this can increase the precision of estimates.
The log-rates for \(k\) populations, \(\boldsymbol \phi_t\), are assigned a multivariate normal distribution (Brandt and Williams 2007): \[\boldsymbol \phi_t \sim \text{Gau}(\boldsymbol \phi_{t-1}, \boldsymbol \Sigma),\] where \(\boldsymbol \Sigma\) is a \(k \times k\) covariance matrix.
The covariance matrix can be decomposed into a diagonal matrix containing scale parameters for each variable, \(\boldsymbol \Delta = diag(\tau_1,\dots \tau_k)\), and a symmetric correlation matrix, \(\boldsymbol \Omega\) (Stan Development Team 2021): \[\boldsymbol \Sigma = \boldsymbol \Delta \boldsymbol \Omega \boldsymbol \Delta\] When the correlation structure is added to the model, then a prior distribution is also required for the correlation matrix. surveil uses the LKJ model, which has a single shape parameter, \(\eta\) (Stan Development Team 2021). If \(\eta=1\), the LKJ model will place uniform prior probability on any \(k \times k\) correlation matrix; as \(\eta\) increases from one, it expresses ever greater skepticism towards large correlations. When \(\eta <1\), the LKJ model becomes ‘concave’—expressing skepticism towards correlations of zero.
If we wanted to add a correlation structure to the model, we would
add cor = TRUE to stan_rw, as in: