Hello everyone,
A few days ago I asked a question about concurvity in a GAM (the
anologue of collinearity in a GLM) implemented in mgcv. I think my
question was a bit unfocussed, so I am retrying again, but with
additional information included about the autocorrelation function. I
have also posted about this on Cross Validated. Given all the model
output, it might make for easier
reading:https://stats.stackexchange.com/questions/577790/high-concurvity-collinearity-between-time-and-temperature-in-gam-predicting-dea
As mentioned previously, I have problems with concurvity in my thesis
research, and don't have access to a statistician who works with time
series, GAMs or R. I'd be very grateful for any (partial) answer,
however short. I'll gladly return the favour where I can! For really
helpful input I'd be more than happy to offer co-authorship on
publication. Deadlines are very close, and I'm heading towards having
no results at all if I can't solve this concurvity issue :(
I'm using GAMs to try to understand the relationship between deaths
and heat-related variables (e.g. temperature and humidity), using
daily time series over a 14-year period from a tropical, low-income
country. My aim is to understand the relationship between these
variables and deaths, rather than pure prediction performance.
The GAMs include distributed lag models (set up as 7-column matrices,
see code at bottom of post), since deaths may occur over several days
following exposure.
Simple GAMs with just time, lagged temperature and lagged
precipitation (a potential confounder) show very high concurvity
between lagged temperature and time, regardless of the many different
ways I have tried to decompose time. The autocorrelation functions
(ACF) however, shows values close to zero, only just about breaching
the 'significance line' in a few instances. It does show patterning
though, although the regularity is difficult to define.
My questions are:
1) Should I be worried about the high concurvity, or can I ignore it
given the mostly non-significant ACF? I've read dozens of
heat-mortality modelling studies and none report on concurvity between
weather variables and time (though one 2012 paper discussed
autocorrelation).
2) If I cannot ignore it, what should I do to resolve it? Would
including an autoregressive term be appropriate, and if so, where can
I find a coded example of how to do this? I've also come across
sequential regression][1]. Would this be more or less appropriate? If
appropriate, a pointer to an example would be really appreciated!
Some example GAMs are specified as follows:
```r
conc38b <- gam(deaths~te(year, month, week, weekday,
bs=c("cr","cc","cc","cc")) + heap +
te(temp_max, lag, k=c(10, 3)) +
te(precip_daily_total, lag, k=c(10, 3)),
data = dat, family = nb, method = 'REML', select = TRUE,
knots = list(month = c(0.5, 12.5), week = c(0.5,
52.5), weekday = c(0, 6.5)))
```
Concurvity for the above model between (temp_max, lag) and (year,
month, week, weekday) is 0.91:
```r
$worst
para te(year,month,week,weekday)
te(temp_max,lag) te(precip_daily_total,lag)
para 1.000000e+00 1.125625e-29
0.3150073 0.6666348
te(year,month,week,weekday) 1.400648e-29 1.000000e+00
0.9060552 0.6652313
te(temp_max,lag) 3.152795e-01 8.998113e-01
1.0000000 0.5781015
te(precip_daily_total,lag) 6.666348e-01 6.652313e-01
0.5805159 1.0000000
```
Output from ```gam.check()```:
```r
Method: REML Optimizer: outer newton
full convergence after 16 iterations.
Gradient range [-0.01467332,0.003096643]
(score 8915.994 & scale 1).
Hessian positive definite, eigenvalue range [5.048053e-05,26.50175].
Model rank = 544 / 544
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.
k' edf k-index p-value
te(year,month,week,weekday) 319.0000 26.6531 0.96 0.06 .
te(temp_max,lag) 29.0000 3.3681 NA NA
te(precip_daily_total,lag) 27.0000 0.0051 NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```
Some output from ```summary(conc38b)```:
```r
Approximate significance of smooth terms:
edf Ref.df Chi.sq p-value
te(year,month,week,weekday) 26.653127 319 166.803 < 2e-16 ***
te(temp_max,lag) 3.368129 27 11.130 0.00145 **
te(precip_daily_total,lag) 0.005104 27 0.002 0.69636
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.839 Deviance explained = 53.3%
-REML = 8916 Scale est. = 1 n = 5107
```
Below are the ACF plots (note limit y-axis = 0.1 for clarity of
pattern). They show peaks at 5 and 15, and then there seems to be a
recurring pattern at multiples of approx. 30 (suggesting month is not
modelled adequately?). Not sure what would cause the spikes at 5 and
15. There is heaping of deaths on the 15th day of each month, to which
deaths with unknown date were allocated. This heaping was modelled
with categorical variable/ factor ```heap``` with 169 levels (0 for
all non-heaping days and 1-168 (i.e. 14 * 12 for each subsequent
heaping day over the 14-year period):
[2]: https://i.stack.imgur.com/FzKyM.png
[3]: https://i.stack.imgur.com/fE3aL.png
I get an identical looking ACF when I decompose time into (year,
month, monthday) as in model conc39 below, although concurvity between
(temp_max, lag) and the time term has now dropped somewhat to 0.83:
```r
conc39 <- gam(deaths~te(year, month, monthday, bs=c("cr","cc","cr")) + heap +
te(temp_max, lag, k=c(10, 4)) +
te(precip_daily_total, lag, k=c(10, 4)),
data = dat, family = nb, method = 'REML', select = TRUE,
knots = list(month = c(0.5, 12.5)))
```
```r
Method: REML Optimizer: outer newton
full convergence after 14 iterations.
Gradient range [-0.001578187,6.155096e-05]
(score 8915.763 & scale 1).
Hessian positive definite, eigenvalue range [1.894391e-05,26.99215].
Model rank = 323 / 323
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.
k' edf k-index p-value
te(year,month,monthday) 79.0000 25.0437 0.93 <2e-16 ***
te(temp_max,lag) 39.0000 4.0875 NA NA
te(precip_daily_total,lag) 36.0000 0.0107 NA NA
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```
Some output from ```summary(conc39)```:
```r
Approximate significance of smooth terms:
edf Ref.df Chi.sq p-value
te(year,month,monthday) 38.75573 99 187.231 < 2e-16 ***
te(temp_max,lag) 4.06437 37 25.927 1.66e-06 ***
te(precip_daily_total,lag) 0.01173 36 0.008 0.557
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.839 Deviance explained = 53.8%
-REML = 8915 Scale est. = 1 n = 5107
```
```r
$worst
para te(year,month,monthday)
te(temp_max,lag) te(precip_daily_total,lag)
para 1.000000e+00 3.261007e-31
0.3313549 0.6666532
te(year,month,monthday) 3.060763e-31 1.000000e+00
0.8266086 0.5670777
te(temp_max,lag) 3.331014e-01 8.225942e-01
1.0000000 0.5840875
te(precip_daily_total,lag) 6.666532e-01 5.670777e-01
0.5939380 1.0000000
```
Modelling time as ```te(year, doy)``` with a cyclic spline for doy and
various choices for k or as ```s(time)``` with various k does not
reduce concurvity either.
The default approach in time series studies of heat-mortality is to
model time with fixed df, generally between 7-10 df per year of data.
I am, however, apprehensive about this approach because a) mortality
profiles vary with locality due to sociodemographic and environmental
characteristics and b) the choice of df is based on higher income
countries (where nearly all these studies have been done) with
different mortality profiles and so may not be appropriate for
tropical, low-income countries.
Although the approach of fixing (high) df does remove more temporal
patterns from the ACF (see model and output below), concurvity between
time and lagged temperature has now risen to 0.99! Moreover,
temperature (which has been a consistent, highly significant predictor
in every model of the tens (hundreds?) I have run, has now turned
non-significant. I am guessing this is because time is now a very
wiggly function that not only models/ removes seasonal variation, but
also some of the day-to-day variation that is needed for the
temperature smooth :
```r
conc20a <- gam(deaths~s(time, k=112, fx=TRUE) + heap +
te(temp_max, lag, k=c(10,3)) +
te(precip_daily_total, lag, k=c(10,3)),
data = dat, family = nb, method = 'REML', select = TRUE)
```
Output from ```gam.check(conc20a, rep = 1000)```:
```r
Method: REML Optimizer: outer newton
full convergence after 9 iterations.
Gradient range [-0.0008983099,9.546022e-05]
(score 8750.13 & scale 1).
Hessian positive definite, eigenvalue range [0.0001420112,15.40832].
Model rank = 336 / 336
Basis dimension (k) checking results. Low p-value (k-index<1) may
indicate that k is too low, especially if edf is close to k'.
k' edf k-index p-value
s(time) 111.0000 111.0000 0.98 0.56
te(temp_max,lag) 29.0000 0.6548 NA NA
te(precip_daily_total,lag) 27.0000 0.0046 NA NA
```
Output from ```concurvity(conc20a, full=FALSE)$worst```:
```r
para s(time) te(temp_max,lag)
te(precip_daily_total,lag)
para 1.000000e+00 2.462064e-19 0.3165236
0.6666348
s(time) 2.462398e-19 1.000000e+00 0.9930674
0.6879284
te(temp_max,lag) 3.170844e-01 9.356384e-01 1.0000000
0.5788711
te(precip_daily_total,lag) 6.666348e-01 6.879284e-01 0.5788381
1.0000000
```
Some output from ```summary(conc20a)```:
```r
Approximate significance of smooth terms:
edf Ref.df Chi.sq p-value
s(time) 1.110e+02 111 419.375 <2e-16 ***
te(temp_max,lag) 6.548e-01 27 0.895 0.249
te(precip_daily_total,lag) 4.598e-03 27 0.002 0.868
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
R-sq.(adj) = 0.843 Deviance explained = 56.1%
-REML = 8750.1 Scale est. = 1 n = 5107
```
ACF functions:
[4]: https://i.stack.imgur.com/7nbXS.png
[5]: https://i.stack.imgur.com/pNnZU.png
Data can be found on my [GitHub][6] site in the file
[data_cross_validated_post2.rds][7]. A csv version is also available.
This is my code:
```r
library(readr)
library(mgcv)
df <- read_rds("data_crossvalidated_post2.rds")
# Create matrices for lagged weather variables (6 day lags) based on
example by Simon Wood
# in his 2017 book ("Generalized additive models: an introduction with
R", p. 349) and
# gamair package documentation
(https://cran.r-project.org/web/packages/gamair/gamair.pdf, p. 54)
lagard <- function(x,n.lag=7) {
n <- length(x); X <- matrix(NA,n,n.lag)
for (i in 1:n.lag) X[i:n,i] <- x[i:n-i+1]
X
}
dat <- list(lag=matrix(0:6,nrow(df),7,byrow=TRUE),
deaths=df$deaths_total,doy=df$doy, year = df$year, month = df$month,
weekday = df$weekday, week = df$week, monthday = df$monthday, time =
df$time, heap=df$heap, heap_bin = df$heap_bin, precip_hourly_dailysum
= df$precip_hourly_dailysum)
dat$temp_max <- lagard(df$temp_max)
dat$temp_min <- lagard(df$temp_min)
dat$temp_mean <- lagard(df$temp_mean)
dat$wbgt_max <- lagard(df$wbgt_max)
dat$wbgt_mean <- lagard(df$wbgt_mean)
dat$wbgt_min <- lagard(df$wbgt_min)
dat$temp_wb_nasa_max <- lagard(df$temp_wb_nasa_max)
dat$sh_mean <- lagard(df$sh_mean)
dat$solar_mean <- lagard(df$solar_mean)
dat$wind2m_mean <- lagard(df$wind2m_mean)
dat$sh_max <- lagard(df$sh_max)
dat$solar_max <- lagard(df$solar_max)
dat$wind2m_max <- lagard(df$wind2m_max)
dat$temp_wb_nasa_mean <- lagard(df$temp_wb_nasa_mean)
dat$precip_hourly_dailysum <- lagard(df$precip_hourly_dailysum)
dat$precip_hourly <- lagard(df$precip_hourly)
dat$precip_daily_total <- lagard( df$precip_daily_total)
dat$temp <- lagard(df$temp)
dat$sh <- lagard(df$sh)
dat$rh <- lagard(df$rh)
dat$solar <- lagard(df$solar)
dat$wind2m <- lagard(df$wind2m)
conc38b <- gam(deaths~te(year, month, week, weekday,
bs=c("cr","cc","cc","cc")) + heap +
te(temp_max, lag, k=c(10, 3)) +
te(precip_daily_total, lag, k=c(10, 3)),
data = dat, family = nb, method = 'REML', select = TRUE,
knots = list(month = c(0.5, 12.5), week = c(0.5,
52.5), weekday = c(0, 6.5)))
conc39 <- gam(deaths~te(year, month, monthday, bs=c("cr","cc","cr")) + heap +
te(temp_max, lag, k=c(10, 4)) +
te(precip_daily_total, lag, k=c(10, 4)),
data = dat, family = nb, method = 'REML', select = TRUE,
knots = list(month = c(0.5, 12.5)))
conc20a <- gam(deaths~s(time, k=112, fx=TRUE) + heap +
te(temp_max, lag, k=c(10,3)) +
te(precip_daily_total, lag, k=c(10,3)),
data = dat, family = nb, method = 'REML', select = TRUE)
```
Thank you if you've read this far!! :-))
[1]:
https://scholar.google.co.uk/scholar?output=instlink&q=info:PKdjq7ZwozEJ:scholar.google.com/&hl=en&as_sdt=0,5&scillfp=17865929886710916120&oi=lle
[2]: https://i.stack.imgur.com/FzKyM.png
[3]: https://i.stack.imgur.com/fE3aL.png
[4]: https://i.stack.imgur.com/7nbXS.png
[5]: https://i.stack.imgur.com/pNnZU.png
[6]: https://github.com/JadeShodan/heat-mortality
[7]:
https://github.com/JadeShodan/heat-mortality/blob/main/data_cross_validated_post2.rds
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