# [R] High concurvity/ collinearity between time and temperature in GAM predicting deaths but low ACF. Does this matter?

j@de@shod@@ m@iii@g oii googiem@ii@com j@de@shod@@ m@iii@g oii googiem@ii@com
Thu Jun 9 17:57:45 CEST 2022

```Hi Simon,

(Sorry, replies and answers are out of sync due to my problems posting
to the list/ messages being held for moderation)

> Did it actually fail, or simply generate warnings?
The model was computed (you're right, not a model failure), but
resulted in warnings (as per previous post) which, frankly, I didn't
understand.

> if I understand your model right there is one free parameter for each observation falling on the 15th
That's right, one observation on each 15th day of the month.

Thank you for the suggestion about the random effects! I had been
wondering about how I could model this heaping with a smooth!

Quick question:

You proposed:

aheap <- heap!="0"
heap + s(heap,bs="re",by=heap) ## fixed mean effect of being a heap
day + a r.e. for each heap day - no effect on non-heap days

Is there a typo in the code above? I don't see the newly created
variable aheap in the model?
Should it read "by = aheap" as follows:

heap + s(heap,bs="re",by=aheap)   ?

So a full model might then look like the one below?

bam0 <- bam(deaths~te(year, month, week, weekday,
bs=c("cr","cc","cc","cc"), k = c(4,5,5,5)) + heap +
s(heap,bs="re",by=aheap)
te(temp_max, lag, k=c(8, 3)) +
te(precip_daily_total, lag, k=c(8, 3)),
data = dat, family = nb, method =
'fREML', select = TRUE, discrete = TRUE,
knots = list(month = c(0.5, 12.5),
week = c(0.5, 52.5), weekday = c(0, 6.5)))

One, hopefully final (!) question:

Is it actually useful at all to keep these observations on the 15th
day of each month (which are huge errors), or am I better off removing
them from the data set (or replacing them with e.g. median values)?
For temperature-mortality modelling it is the day-to-day variation in
deaths and temperature that is of interest. So is modelling heaping
actually useful at all, given that this variable changes on a monthly
basis? (I think you are alluding to this, but I just want to make
sure).

If I take them out altogether, would I be best off removing all data
for these dates, so that the time series jumps from day 14 to day 16?
Or would this create problems with e.g. the distributed lag model?

Sorry for all these questions! Have been struggling with this for
months (posted on Cross Validated about the heaping issue too), and
feel I am finally getting somewhere with your help!

On Thu, 9 Jun 2022 at 15:24, Simon Wood <simon.wood using bath.edu> wrote:
>
>
> > Hi Simon,
> >
> > Thanks so much for this!! (Apologies if this is a double posting. I
> > seem to have a problem getting messages through to the list).
> >
> > I have two follow up questions, if you don't mind.
> >
> > 1. Does including an autoregressive term not adjust away part of the
> > effect of the response in a distributed lag model (where the outcome
> > accumulates over time)?
>
> - the hope is that it approximately deals with short timescale stuff
> without interfering with the longer timescales to the same extent as a
> high rank smooth would.
>
> > 2. I've tried to fit a model using bam (just a first attempt without
> > AR term), but including the factor variable heap creates errors:
> >
> > bam0 <- bam(deaths~te(year, month, week, weekday,
> > bs=c("cr","cc","cc","cc"), k = c(4,5,5,5)) + heap +
> >                                       te(temp_max, lag, k=c(8, 3)) +
> >                                       te(precip_daily_total, lag, k=c(8, 3)),
> >                                      data = dat, family = nb, method =
> > 'fREML', select = TRUE, discrete = TRUE,
> >                                      knots = list(month = c(0.5, 12.5),
> > week = c(0.5, 52.5), weekday = c(0, 6.5)))
> >
> > This model results in errors:
> >
> > Warning in estimate.theta(theta, family, y, mu, scale = scale1, wt = G\$w,  :
> >    step failure in theta estimation
> > Warning in sqrt(family\$dev.resids(object\$y, object\$fitted.values,
> > object\$prior.weights)) :
> >    NaNs produced
> >
> Did it actually fail, or simply generate warnings?
>
> 'bam' handles factors, but if I understand your model right there is one
> free parameter for each observation falling on the 15th, so that you
> will fit data for those days exactly (and might just as well have
> dropped them, for all the information they contribute to the rest of the
> model). If you want a structure like this, I'd be inclined to make the
> heap variable random, something like...
>
> aheap <- heap!="0"
>
> heap + s(heap,bs="re",by=heap) ## fixed mean effect of being a heap day
> + a r.e. for each heap day - no effect on non-heap days
>
> best,
>
> Simon
>
> > Including heap as as.numeric(heap) runs the model without error
> > messages or warnings, but model diagnostics look terrible, and it also
> > doesn't make sense (to me) to make heap a numeric. The factor variable
> > heap (with 169 levels) codes the fact that all deaths for which no
> > date was known, were registered on the 15th day of each month. I've
> > coded all non-heaping days as 0. All heaping days were coded as a
> > value between 1-168. The time series spans 14 years, so a heaping day
> > in each month results in 14*12 levels = 168, plus one level for
> > non-heaping days.
> >
> > So my second question is: Does bam allow factor variables? And if not,
> > how should I model this heaping on the 15th day of the month instead?
> >
> > With thanks,
> >
> >
> > On Wed, 8 Jun 2022 at 12:05, Simon Wood <simon.wood using bath.edu> wrote:
> >> I would not worry too much about high concurvity between variables like
> >> temperature and time. This just reflects the fact that temperature has a
> >> strong temporal pattern.
> >>
> >> I would also not be too worried about the low p-values on the k check.
> >> The check only looks for pattern in the residuals when they are ordered
> >> with respect to the variables of a smooth. When you have time series
> >> data and some smooths involve time then it's hard not to pick up some
> >> degree of residual auto-correlation, which you often would not want to
> >> model with a higher rank smoother.
> >>
> >> The NAs  for the distributed lag terms just reflect the fact that there
> >> is no obvious way to order the residuals w.r.t. the covariates for such
> >> terms, so the simple check for residual pattern is not really possible.
> >>
> >> One simple approach is to fit using bam(...,discrete=TRUE) which will
> >> let you specify an AR1 parameter to mop up some of the residual
> >> auto-correlation without resorting to a high rank smooth that then does
> >> all the work of the covariates as well. The AR1 parameter can be set by
> >> looking at the ACF of the residuals of the model without this. You need
> >> to look at the ACF of suitably standardized residuals to check how well
> >> this has worked.
> >>
> >> best,
> >>
> >> Simon
> >>
> >> On 05/06/2022 20:01, jade.shodan--- via R-help wrote:
> >>> 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
> >>> output, it might make for easier
> >>>
> >>> 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.
> >>> (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.
> >>> (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.
> >>> 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.
> >>> (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(mgcv)
> >>>
> >>>
> >>> # 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!! :-))
> >>>
> >>>     [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
> >>>
> >>> ______________________________________________
> >>> R-help using r-project.org mailing list -- To UNSUBSCRIBE and more, see
> >>> https://stat.ethz.ch/mailman/listinfo/r-help
> >>> and provide commented, minimal, self-contained, reproducible code.
> >> --
> >> Simon Wood, School of Mathematics, University of Edinburgh,
> >> https://www.maths.ed.ac.uk/~swood34/
> >>
> --
> Simon Wood, School of Mathematics, University of Edinburgh,
> https://www.maths.ed.ac.uk/~swood34/
>

```