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, Hughes, and
Lee (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, yt, conditional on a given level of risk, eϕt, and known population at risk, pt, is: yt ∼ Pois(pt ⋅ eϕt) where t indexes the time period.
Next, we need a model for the log-rates, ϕt. 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: ϕt ∼ Gau(ϕt − 1, τ2) 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 ϕ1. For this prior, surveil uses a normal distribution. The scale parameter, τ, 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: yt ∼ Binom(pt ⋅ g−1(ϕt)) 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 2e-05 seconds
#> Chain 1: 1000 transitions using 10 leapfrog steps per transition would take 0.2 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: 0.259 seconds (Warm-up)
#> Chain 1: 0.147 seconds (Sampling)
#> Chain 1: 0.406 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: 0.242 seconds (Warm-up)
#> Chain 2: 0.146 seconds (Sampling)
#> Chain 2: 0.388 seconds (Total)
#> Chain 2:
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 3).
#> Chain 3:
#> Chain 3: Gradient evaluation took 1.3e-05 seconds
#> Chain 3: 1000 transitions using 10 leapfrog steps per transition would take 0.13 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: 0.255 seconds (Warm-up)
#> Chain 3: 0.167 seconds (Sampling)
#> Chain 3: 0.422 seconds (Total)
#> Chain 3:
#>
#> SAMPLING FOR MODEL 'RW' NOW (CHAIN 4).
#> Chain 4:
#> Chain 4: Gradient evaluation took 1.3e-05 seconds
#> Chain 4: 1000 transitions using 10 leapfrog steps per transition would take 0.13 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: 0.254 seconds (Warm-up)
#> Chain 4: 0.147 seconds (Sampling)
#> Chain 4: 0.401 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 170.02665 157.97857 182.86964
#> 2 2000 Black or African American 166.33487 156.11270 176.38981
#> 3 2001 Black or African American 168.26455 158.90022 178.92890
#> 4 2002 Black or African American 169.02493 159.99398 178.50573
#> 5 2003 Black or African American 167.14335 157.75147 177.21097
#> 6 2004 Black or African American 166.99334 157.62439 177.29462
#> 7 2005 Black or African American 159.07992 150.37551 167.57004
#> 8 2006 Black or African American 154.69952 146.30444 163.08918
#> 9 2007 Black or African American 152.82305 144.05898 161.36941
#> 10 2008 Black or African American 149.88290 141.56708 158.48908
#> 11 2009 Black or African American 143.51200 136.01516 152.06417
#> 12 2010 Black or African American 138.95339 131.74345 146.58867
#> 13 2011 Black or African American 130.99578 124.04188 138.24500
#> 14 2012 Black or African American 125.26712 118.29093 131.76739
#> 15 2013 Black or African American 124.87700 118.37905 131.15280
#> 16 2014 Black or African American 126.31612 120.08933 133.34435
#> 17 2015 Black or African American 122.42635 115.90959 128.81236
#> 18 2016 Black or African American 122.01154 115.81850 128.22206
#> 19 2017 Black or African American 122.56087 115.53571 129.62014
#> 20 1999 Hispanic 101.55753 94.66497 108.41190
#> 21 2000 Hispanic 104.12220 98.12087 111.05781
#> 22 2001 Hispanic 102.45516 97.16257 108.23882
#> 23 2002 Hispanic 101.49065 96.12717 106.92772
#> 24 2003 Hispanic 100.55043 95.24610 106.11101
#> 25 2004 Hispanic 100.93174 95.77989 106.49874
#> 26 2005 Hispanic 98.62223 93.89012 103.67906
#> 27 2006 Hispanic 96.89503 92.22127 101.78652
#> 28 2007 Hispanic 94.20343 89.99999 98.88811
#> 29 2008 Hispanic 90.53801 85.91908 94.86160
#> 30 2009 Hispanic 88.82908 84.39509 93.09947
#> 31 2010 Hispanic 88.00658 83.96580 92.08002
#> 32 2011 Hispanic 87.44804 83.73633 91.46629
#> 33 2012 Hispanic 87.83121 83.82511 91.78439
#> 34 2013 Hispanic 87.25406 83.51662 91.06155
#> 35 2014 Hispanic 85.81301 82.05005 89.73375
#> 36 2015 Hispanic 85.79696 82.08703 89.46903
#> 37 2016 Hispanic 84.77330 81.10755 88.25857
#> 38 2017 Hispanic 85.85186 81.96673 89.83291
#> 39 1999 White 135.20228 130.22405 140.34962
#> 40 2000 White 136.49513 131.75820 141.47613
#> 41 2001 White 134.08293 129.76189 138.62881
#> 42 2002 White 130.16819 125.96077 134.54539
#> 43 2003 White 127.49020 123.21074 131.85411
#> 44 2004 White 120.01979 115.85622 124.13146
#> 45 2005 White 115.05855 111.17396 119.17619
#> 46 2006 White 109.70595 106.04387 113.41961
#> 47 2007 White 108.19638 104.42870 112.05343
#> 48 2008 White 103.46708 99.92724 107.05635
#> 49 2009 White 98.90905 95.70196 102.23229
#> 50 2010 White 96.27780 92.90287 99.61640
#> 51 2011 White 93.63307 90.49818 96.82609
#> 52 2012 White 91.11946 88.10232 94.25843
#> 53 2013 White 90.10576 86.85408 93.38720
#> 54 2014 White 91.42235 88.30207 94.65148
#> 55 2015 White 93.13817 90.07991 96.29863
#> 56 2016 White 89.08309 86.10363 92.05630
#> 57 2017 White 92.33807 89.16341 95.72758This 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 0.001700267 0.001579786 0.001828696 Black or African American 1999 471
#> 2 2000 0.001663349 0.001561127 0.001763898 Black or African American 2000 455
#> 3 2001 0.001682646 0.001589002 0.001789289 Black or African American 2001 505
#> 4 2002 0.001690249 0.001599940 0.001785057 Black or African American 2002 539
#> 5 2003 0.001671434 0.001577515 0.001772110 Black or African American 2003 546
#> 6 2004 0.001669933 0.001576244 0.001772946 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 0.00000000 0.000000 0.000000
#> 2 2000 Black or African American -2.08947327 -9.173061 4.921481
#> 3 2001 Black or African American 1.22292389 -5.228113 8.517048
#> 4 2002 Black or African American 0.51375009 -5.678609 7.227592
#> 5 2003 Black or African American -1.06615263 -7.119626 5.249154
#> 6 2004 Black or African American -0.03761518 -6.128720 6.832813Those 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 0.000000 0.000000 0.000000
#> 2 2000 Black or African American -2.089473 -9.173061 4.921481
#> 3 2001 Black or African American -0.926500 -9.128808 7.401318
#> 4 2002 Black or African American -0.465462 -9.008875 8.659555
#> 5 2003 Black or African American -1.570460 -10.346213 7.649895
#> 6 2004 Black or African American -1.659250 -9.779427 7.734596Each 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 ± .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, ϕt, are assigned a multivariate normal distribution (Brandt and Williams 2007): ϕt ∼ Gau(ϕt − 1, Σ), where Σ is a k × k covariance matrix.
The covariance matrix can be decomposed into a diagonal matrix containing scale parameters for each variable, Δ = diag(τ1, …τk), and a symmetric correlation matrix, Ω (Stan Development Team 2021): Σ = ΔΩΔ 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, η (Stan Development Team 2021). If η = 1, the LKJ model will place uniform prior probability on any k × k correlation matrix; as η increases from one, it expresses ever greater skepticism towards large correlations. When η < 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: