SMHS TimeSeriesAnalysis LOS

Time series regression studies in environmental epidemiology (London Ozone Study 2002-2006)

A time series regression analysis of a London ozone dataset including daily observations from 1 January 2002 to 31 December 2006. Each day has records of (mean) ozone levels that day, and the total number of deaths that occurred in the city.

Questions

Is there an association between day-to-day variation in ozone levels and daily risk of death?

Is ozone exposure associated with the outcome is death or other confounders - temperature and relative humidity?

Reference: Bhaskaran K, Gasparrini A, Hajat S, Smeeth L, Armstrong B. Time series regression studies in environmental epidemiology. International Journal of Epidemiology. 2013;42(4):1187-1195. doi:10.1093/ije/dyt092. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3780998/

Load the Data

library(foreign)
#07_LondonOzonPolutionData_2006_TS.csv
#data <- read.csv("https://umich.instructure.com/files/720873/download?download_frd=1")
data <- read.dta("https://umich.instructure.com/files/721042/download?download_frd=1")

#Set the Default Action for Missing Data to na.exclude
options(na.action="na.exclude")

Exploratory Analyses

#set the plotting parameters for the plot

oldpar <- par(no.readonly=TRUE)
par(mex=0.8,mfrow=c(2,1))

#sub-plot for daily deaths, with vertical lines defining years

plot(data$\$$date,data$\$$numdeaths,pch=".",main="Daily deaths over time",  
   ylab="Daily number of deaths",xlab="Date")
abline(v=data$\$$date[grep("-01-01",data$\$$date)],col=grey(0.6),lty=2)

#plot for ozone levels

plot(data$\$$date,data$\$$ozone,pch=".",main="Ozone levels over time",
    ylab="Daily mean ozone level(ug/m3)",xlab="Date")
abline(v=data$\$$date[grep("-01-01",data$\$$date)],col=grey(0.6),lty=2)
par(oldpar)
layout(1)

#descriptive statistics

summary(data)

#correlations

cor(data[,2:4])
#scale exposure
data$\$$ozone10 <- data$\$$ozone/10

Modelling Seasonality and Long-Term Trend

#option 1: time-stratified model
#generate month and year

data$\$$month <- as.factor(months(data$\$$date,abbr=TRUE))
data$\$$year <- as.factor(substr(data$\$$date,1,4))

#fit a Poisson model with a stratum for each month nested in year
#(use of quasi-Poisson family for scaling the standard errors)

model1 <- glm(numdeaths ~ month/year,data,family=quasipoisson) 
summary(model1)

#compute predicted number of deaths from this model

pred1 <- predict(model1,type="response")

#Figure 2a: Three alternative ways of modelling long-term patterns in the data (seasonality and trends)

plot(data$\$$date,data$\$$numdeaths,ylim=c(100,300),pch=19,cex=0.2,col=grey(0.6),
    main="Time-stratified model (month strata)",ylab="Daily number of deaths", xlab="Date")
lines(data$\$$date, pred1,lwd=2)

#Option 2: periodic functions model (fourier terms)<br>
#use function harmonic, in package '''tsModel''' 

 install.packages("tsModel"); library(tsModel)

#4 sine-cosine pairs representing different harmonics with period 1 year

 data$\$$time <- seq(nrow(data))
fourier <- harmonic(data$\$$time,nfreq=4,period=365.25)

#fit a Poisson model Fourier terms + linear term for trend <br>
#(use of quasi-Poisson family for scaling the standard errors)

 model2 <- glm(numdeaths ~ fourier +time,data,family=quasipoisson) 
 summary(model2)

#compute predicted number of deaths from this model

 pred2 <- predict(model2,type="response")

#Figure 2b

 plot(data$\$$date, data$\$$numdeaths,ylim=c(100,300),pch=19,cex=0.2,col=grey(0.6),
     main="Sine-cosine functions (Fourier terms)",ylab="Daily number of deaths", xlab="Date")
 lines(data$\$$date, pred2,lwd=2)

#Option 3: Spline Model: Flexible Spline Functions
#generate spline terms, use function bs in package splines

library(splines)

#A CUBIC B-SPLINE WITH 32 EQUALLY-SPACED KNOTS + 2 BOUNDARY KNOTS
#Note: the 35 basis variables are set as df, with default knots placement. see ?bs
#other types of splines can be produced with the function ns. see ?ns

spl <- bs(data$\$$time,degree=3,df=35)<br>
#Fit a Poisson Model Fourier Terms + Linear Term for Trend

 model3 <- glm(numdeaths ~ spl,data,family=quasipoisson)
 summary(model3)

#compute predicted number of deaths from this model

 pred3 <- predict(model3,type="response")

#FIGURE 2C

 plot(data$\$$date,data$\$$numdeaths,ylim=c(100,300),pch=19,cex=0.2,col=grey(0.6), 
     main="Flexible cubic spline model",ylab="Daily number of deaths", xlab="Date")
 lines(data$\$$date,pred3,lwd=2)

Plot Response Residuals Over Time From Model 3

#GENERATE RESIDUALS

res3 <- residuals(model3,type="response")

#Figure 3: Residual variation in daily deaths after ‘removing’ (i.e. modelling) season and long-term trend.

plot(data$\$$date,res3,ylim=c(-50,150),pch=19,cex=0.4,col=grey(0.6),
     main="Residuals over time",ylab="Residuals (observed-fitted)",xlab="Date")
 abline(h=1,lty=2,lwd=2)

<b>Estimate ozone-mortality association - controlling for confounders</b>

#compare the RR (and CI using '''ci.lin''' in package '''Epi''')

 install.packages("Epi"); library(Epi)

#unadjusted model

 model4 <- glm(numdeaths ~ ozone10,data,family=quasipoisson)
 summary(model4)
 (eff4 <- ci.lin(model4,subset="ozone10",Exp=T))

#control for seasonality (with spline as in model 3)

 model5 <- update(model4, .~. + spl)
 summary(model5)
 (eff5 <- ci.lin(model5,subset="ozone10",Exp=T))

#control for temperature - temperature modelled with categorical variables for deciles
 
 cutoffs <- quantile(data$\$$temperature,probs=0:10/10)
tempdecile <- cut(data$\$$temperature,breaks=cutoffs,include.lowest=TRUE)
 model6 <- update(model5,.~.+tempdecile)
 summary(model6)
 (eff6 <- ci.lin(model6,subset="ozone10",Exp=T))

<b>Build a summary table with effect as percent increase</b>

 tabeff <- rbind(eff4,eff5,eff6)[,5:7]
 tabeff <- (tabeff-1)*100
 dimnames(tabeff) <- list(c("Unadjusted","Plus season/trend","Plus temperature"), c("RR","ci.low","ci.hi"))
 round(tabeff,2)

#explore the lagged (delayed) effects

#SINGLE-LAG MODELS

#prepare the table with estimates

 tablag <- matrix(NA,7+1,3,dimnames=list(paste("Lag",0:7),  c("RR","ci.low","ci.hi")))

#iterate

 for(i in 0:7) {
     #lag ozone and temperature variables
     ozone10lag <- Lag(data$\$$ozone10,i)
    tempdecilelag <- cut(Lag(data$\$$temperature,i),breaks=cutoffs,   include.lowest=TRUE)

#define the transformation for temperature

#lag same as above, but with strata terms instead than linear
 
     mod <- glm(numdeaths ~ ozone10lag + tempdecilelag + spl,data,   family=quasipoisson)
     tablag[i+1,] <- ci.lin(mod,subset="ozone10lag",Exp=T)[5:7]</blockquote>
 }
 tablag

#Figure 4A: Modelling lagged (delayed) associations between ozone exposure and survival/death outcome.

 plot(0:7,0:7,type="n",ylim=c(0.99,1.03),main="Lag terms modelled one at a time", xlab="Lag (days)",
     ylab="RR and 95%CI per 10ug/m3 ozone increase")</blockquote>
 abline(h=1)
 arrows(0:7,tablag[,2],0:7,tablag[,3],length=0.05,angle=90,code=3)
 points(0:7,tablag[,1],pch=19)

<b>Model Checking</b>

#generate deviance residuals from unconstrained distributed lag model

 res6 <- residuals(model6,type="deviance")

#Figure A1: Plot of deviance residuals over time (London data)

 plot(data$\$$date,res6,ylim=c(-5,10),pch=19,cex=0.7,col=grey(0.6),
     main="Residuals over time",ylab="Deviance residuals",xlab="Date")
abline(h=0,lty=2,lwd=2)

#Figure A2a: Residual plot for Model6: the residuals relate to the unconstrained distributed lag model with ozone

#(lag days 0 to 7 inclusive), adjusted for temperature at the same lags. The spike in the plot of residuals relate to

#the 2003 European heat wave, and indicate that the current model does not explain the data over this period well.

pacf(res6,na.action=na.omit,main="From original model")

#Include the 1-Day Lagged Residual in the Model

model9 <- update(model6,.~.+Lag(res6,1))

#Figure A2b: residuals related to the unconstrained distributed lag model with ozone (lag days 0 to 7 inclusive),

#adjusted for temperature at the same lags

pacf(residuals(model9,type="deviance"),na.action=na.omit,   
    main="From model adjusted for residual autocorrelation")

Irish Longitudinal Study on Ageing Example

The Irish Longitudinal Study on Ageing (TILDA), 2009-2011

http://www.icpsr.umich.edu/icpsrweb/ICPSR/studies/34315

Kenny, Rose Anne. The Irish Longitudinal Study on Ageing (TILDA),

2009-2011. ICPSR34315-v1. Ann Arbor, MI: Inter-university Consortium

Bibliographic Citation: for Political and Social Research [distributor], 2014-07-16.

http://doi.org/10.3886/ICPSR34315.v1

The Irish Longitudinal Study on Ageing (TILDA) is a major inter-institutional initiative led by Trinity College, Dublin, to improve in the quantity and quality of data, research and information related to aging in Ireland. Eligible respondents for this study include individuals aged ≥ 50 and their spouses or partners of any age. Annual interviews on a two yearly basis (N=8,504 people) in Ireland, collecting detailed information on all aspects of their lives, including the economic (pensions, employment, living standards), health (physical, mental, service needs and usage) and social aspects (contact with friends and kin, formal and informal care, social participation). Survey interviews, physical, and biological data are collected along with demographic variables (e.g., age, sex, marital status, household composition, education, and employment), and activities of daily living (ADL), aging, childhood, depression (psychology), education, employment, exercise, eyesight, families, family life, etc.

# download the RDA data object (ICPSR_34315.zip)
# load in the data into RStudio
dataURL <- "https://umich.instructure.com/files/703606/download?download_frd=1"
load(url(dataURL))
head(da34315.0001); data_colnames <- colnames(da34315.0001)
vars <- da34315.0001

vars; head(vars); summary(vars); data_colnames

[1]	”ID"	”HOUSEHOLD”
[3]	"CLUSTER"	"STRATUM"
[5]	”REGION”	"CAPIWEIGHT"
[7]	"IN_SCQ"	"SCQ_WEIGHT"
[9]	"AGE"	"SEX"
[11]	"NML"	"CM003"
...
[1673]	"HA_WEIGHT"	"IN_HA"
[1675]	"SR_HEIGHT_CENTIMETRES"	"HEIGHT"
[1677]	"SR_WEIGHT_KILOGRAMMES"	"WEIGHT"
[1679]	"COGMMSE"	"FRGRIPSTRENGTHD"
[1681]	"FRGRIPSTRENGTHND"	"VISUALACUITYLEFT"
[1683]	"VISUALACUITYRIGHT"	"BPSEATEDSYSTOLIC1"
[1685]	"BPSEATEDSYSTOLIC2"	"BPSEATEDDIASTOLIC1"
[1687]	"BPSEATEDDIASTOLIC2"	"BPSEATEDSYSTOLICMEAN"
[1689]	"BPSEATEDDIASTOLICMEAN"	"BPHYPERTENSION"
[1691]	"FRBMI"	"FRWAIST"
[1693]	"FRHIP"	"FRWHR"
[1695]	"WEARGLASSES"	"WOREGLASSESDURINGTEST"
[1697]	"BLOODS_CHOL"	"BLOODS_HDL"
[1699]	"BLOODS_LDL"	"BLOODS_TRIG"
[1701]	"BLOODS_TIMEBETWEENLASTMEALANDASS"	"DELAY_HA"
[1703]	"PICMEMSCORE"	"PICRECALLSCORE"
[1705]	"PICRECOGSCORE"	"VISREASONING"
[1707]	"GRIPTEST1D"	"GRIPTEST2D"
[1709]	"GRIPTEST1ND"	"GRIPTEST2ND"
[1711]	"GRIPTESTDOMINANT"	"GRIPTESTSITTING"
[1713]	"TEMPERATURE"	"SCQSOCACT1"
...
[1981]	"SOCPROXCHLD4"	"SCRFLU"
[1983]	"SCRCHOL"	"SCRPROSTATE"
[1985]	"SCRBREASTLUMPS"	"SCRMAMMOGRAM"
[1987]	"BEHALC_FREQ_WEEK"	"BEHALC_DRINKSPERDAY"
[1989]	"BEHALC_DRINKSPERWEEK"	"BEHALC_DOH_LIMIT"
[1991]	"BEHSMOKER"	"BEHCAGE"

# extract some data elements
df1 <- data.frame(vars)

df_Irish_small <- df1[, c("ID",  "HOUSEHOLD",  "AGE", "SEX"  , "HA_WEIGHT", "HEIGHT" ,                         
   "WEIGHT", "COGMMSE", "FRGRIPSTRENGTHD", "VISUALACUITYLEFT",       
   "VISUALACUITYRIGHT",    "BPSEATEDSYSTOLIC1",      
   "BPSEATEDSYSTOLIC2",       "BPSEATEDDIASTOLIC1",     
   "BPSEATEDDIASTOLIC2",      "BPSEATEDSYSTOLICMEAN",   
   "BPSEATEDDIASTOLICMEAN",   "BPHYPERTENSION",
   "WEARGLASSES",   "WOREGLASSESDURINGTEST",  
   "BLOODS_CHOL",   "BLOODS_HDL",    
   "BLOODS_LDL",  "BLOODS_TRIG",   
   "PICMEMSCORE",   "PICRECALLSCORE",
   "PICRECOGSCORE", "VISREASONING",  
   "TEMPERATURE",  "SOCPROXCHLD4",   "SCRFLU", "SCRCHOL",        "SCRPROSTATE",   
   "SCRBREASTLUMPS", "SCRMAMMOGRAM",  
   "BEHALC_FREQ_WEEK",        "BEHALC_DRINKSPERDAY",    
   "BEHALC_DRINKSPERWEEK",    "BEHALC_DOH_LIMIT",       
   "BEHSMOKER",      "BEHCAGE" )
   ]

summary(df_Irish_small); head(df_Irish_small)
write.table(df_Irish_small , "data.csv", sep=",")

Applications

Frailty associations with sustained attention measures⁵

Multinomial logistic regression analyses were used to examine frailty as the outcome variable were performed to determine associations between the sustained attention measures and prefrailty or frailty. Binary logistic regression analyses determined significant associations between the sustained attention measures and the individual frailty components. The regression models included age and gender and were also extended to include additional measures of cognitive processing speed (cognitive RT from CRT), executive function (Delta CTT), number of chronic conditions, and number of medications. We also included the quadratic term age2 to allow for any potential nonlinear effects of age on frailty in each regression model. For the independent variables in the multinomial logistic regression models, relative risk (RR) ratios with 95% confidence intervals (CIs) were provided. For the independent variables in the binary logistic regression models, OR with 95% CI were provided.

Multivariable logistic regression examining the association between social relationships and depression, anxiety, and suicidal ideation⁶

⁵http://psychsocgerontology.oxfordjournals.org/content/early/2013/03/13/geronb.gbt009.full

⁶http://www.jad-journal.com/article/S0165-0327%2815%2900145-7/fulltext

See also

• Previous Section on Time-series analysis

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• SOCR Home page: http://www.socr.ucla.edu

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