Exercise 7. Model cause-specific mortality with Poisson regression
In this exercise we model, using Poisson regression, cause-specific mortality of patients diagnosed with localised (stage==1) melanoma.
In exercise 9 we model cause-specific mortality using Cox regression and in exercise 28 we use flexible parametric models. The aim is to illustrate that these three methods are very similar.
The aim of these exercises is to explore the similarities and differences to these three approaches to modelling. We will be comparing the results (and their interpretation) as we proceed through the exercises.
Load the diet data using time-on-study as the timescale.
You may have to install the required packages the first time you use them. You can install a package by install.packages("package_of_interest") for each package you require.
library(biostat3)
library(dplyr) # for data manipulation
library(car) # for car::linearHypothesis -> biostat3::lincomLoad the melanoma data and explore it.
## Read melanoma data
## Create a new dataset with only localised cancer
melanoma.l <- filter(biostat3::melanoma, stage=="Localised")
head( melanoma.l )
summary(melanoma.l)Rates can be modelled on different timescales, e.g., attained age, time-since-entry, calendar time. Plot the CHD incidence rates both by attained age and by time-since-entry. Is there a difference? Do the same for CHD hazard by different energy intakes (hieng).
(a)
i.
## Plot Kaplan-Meier curve using survfit
## Create a new event indicator
melanoma.l <- mutate(melanoma.l,
death_cancer = as.numeric(status=="Dead: cancer") )
## Create a fitted object for our subcohort
## using survfit
sfit7a1 <- survfit(Surv(surv_mm, event=death_cancer) ~ year8594,
data = melanoma.l )
## Have a look at the fitted object
str(sfit7a1, 1)## List of 17
## $ n : int [1:2] 2145 3173
## $ time : num [1:381] 0.5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 ...
## $ n.risk : num [1:381] 2145 2143 2140 2138 2135 ...
## $ n.event : num [1:381] 0 3 0 2 2 0 4 5 4 5 ...
## $ n.censor : num [1:381] 2 0 2 1 3 2 3 2 5 3 ...
## $ surv : num [1:381] 1 0.999 0.999 0.998 0.997 ...
## $ std.err : num [1:381] 0 0.000809 0.000809 0.001045 0.001237 ...
## $ cumhaz : num [1:381] 0 0.0014 0.0014 0.00234 0.00327 ...
## $ std.chaz : num [1:381] 0 0.000808 0.000808 0.001044 0.001237 ...
## $ strata : Named int [1:2] 249 132
## ..- attr(*, "names")= chr [1:2] "year8594=Diagnosed 75-84" "year8594=Diagnosed 85-94"
## $ type : chr "right"
## $ logse : logi TRUE
## $ conf.int : num 0.95
## $ conf.type: chr "log"
## $ lower : num [1:381] 1 0.997 0.997 0.996 0.994 ...
## $ upper : num [1:381] 1 1 1 1 0.999 ...
## $ call : language survfit(formula = Surv(surv_mm, event = death_cancer) ~ year8594, data = melanoma.l)
## - attr(*, "class")= chr "survfit"## Plot the survival curve (with some bells and whistles)
plot(sfit7a1,
## No automatic labelling of the curve (we do that ourselves)
mark.time=F,
## Time is measured in months, but we want to see it in years
xscale=12,
## Make the plot prettier
xlab="Years since diagnosis",
ylab="S(t)",
col=c("blue","red"),
lty=c("solid","dashed"))
## Add legend too
legend("bottomleft",legend=levels(melanoma.l$year8594),col=c("blue","red"),lty=c("solid","dashed"), bty="n")
### TRY IF YOU WANT ###
if (FALSE) {
library(survMisc)
## Note: `autoplot(sfit7a1)` was broken; I have submitted a pull request to fix this
## autoplot(sfit7a1)
## alternatively:
autoplot(sfit7a1, timeTicks = "custom", times= seq(0, 20*12, 5*12))
}Survival is better during the latter period.
ii.
## To plot smoothed hazards, we use the muhaz package (using the muhaz2 wrapper)
plot(muhaz2(Surv(surv_mm/12, status == "Dead: cancer") ~ year8594, data=melanoma.l),
xlab="Years since diagnosis", col=c("blue","red"), lty=1:2)
Mortality is lower during the latter period.
iii.
## Compare with Kaplan-Meier plot
par(mfrow=c(1,2)) ## Two graphs in the same window
plot(sfit7a1,
## No automatic labelling of the curve (we do that ourselves)
mark.time=F,
## Time is measured in months, but we want to see it in years
xscale=12,
## Make the plot prettier
xlab="Years since diagnosis",
ylab="S(t)",
col=c("blue","red"),
lty=c("solid","dashed"))
plot(muhaz2(Surv(surv_mm/12, status == "Dead: cancer") ~ year8594, data=melanoma.l),
xlab="Years since diagnosis", col=c("blue","red"),lty=c("solid","dashed"))
The two graphs both show that prognosis is better during the latter period. Patients diagnosed during the latter period have lower mortality and higher survival.
(b)
## year8594 tstop event rate lower
## year8594=Diagnosed 75-84 Diagnosed 75-84 271953.5 572 0.002103301 0.001934444
## year8594=Diagnosed 85-94 Diagnosed 85-94 191565.5 441 0.002302085 0.002092214
## upper
## year8594=Diagnosed 75-84 0.002282950
## year8594=Diagnosed 85-94 0.002527306The estimated mortality rate is lower for patients diagnosed during the early period. This is not consistent with what we saw in previous analyses. The inconsistency is due to the fact that we have not controlled for time since diagnosis. look at the graph of the estimated hazards (on the previous page) and try and estimate the overall average value for each group. We see that the average hazard for patients diagnosed in the early period is drawn down by the low mortality experienced by patients 10 years subsequent to diagnosis.
(c)
i.
## Calculate the incidence rate by time of diagnosis
## but with new variables
melanoma.l2 <- mutate(melanoma.l,
## Update the death indicator (only count deaths within 120 months)
## death_cancer = death_cancer * as.numeric(surv_mm<=120),
death_cancer = ifelse(surv_mm<=120 & status == "Dead: cancer",1,0),
## Create a new time variable
## surv_mm = pmin(surv_mm, 120)
surv_mm = ifelse(surv_mm<=120, surv_mm, 120)
)
## Calculate the rates on the truncated data
rates_by_diag_yr2 <- survRate(Surv(surv_mm, death_cancer) ~ year8594, data=melanoma.l2)
rates_by_diag_yr2## year8594 tstop event rate lower
## year8594=Diagnosed 75-84 Diagnosed 75-84 198012.5 519 0.002621047 0.002400371
## year8594=Diagnosed 85-94 Diagnosed 85-94 190507.5 441 0.002314869 0.002103833
## upper
## year8594=Diagnosed 75-84 0.002856555
## year8594=Diagnosed 85-94 0.002541342Now that we have restricted follow-up to a maximum of 10 years we see that the average mortality rate for patients diagnosed in the early period is higher than for the latter period. This is consistent with the graphs we examined in part (a).
ii.
## [1] 0.8831852##
## Comparison of Poisson rates
##
## data: event time base: tstop
## count1 = 441, expected count1 = 470.73, p-value = 0.05682
## alternative hypothesis: true rate ratio is not equal to 1
## 95 percent confidence interval:
## 0.7761294 1.0046616
## sample estimates:
## rate ratio
## 0.8831852iii.
## Use glm to estimate the rate ratios
## we scale the offset term to 1000 person-years
poisson7c <- glm( death_cancer ~ year8594 + offset( log( surv_mm/12/1000 ) ), family=poisson, data=melanoma.l2 )
summary( poisson7c )##
## Call:
## glm(formula = death_cancer ~ year8594 + offset(log(surv_mm/12/1000)),
## family = poisson, data = melanoma.l2)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.7931 -0.7568 -0.5632 -0.3081 3.0550
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.44848 0.04389 78.566 <2e-16 ***
## year8594Diagnosed 85-94 -0.12422 0.06476 -1.918 0.0551 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 4814.3 on 5317 degrees of freedom
## Residual deviance: 4810.6 on 5316 degrees of freedom
## AIC: 6734.6
##
## Number of Fisher Scoring iterations: 6## also for collapsed data
summary(glm( event ~ year8594 + offset( log( tstop/12/1000 ) ), family=poisson, data=rates_by_diag_yr2))##
## Call:
## glm(formula = event ~ year8594 + offset(log(tstop/12/1000)),
## family = poisson, data = rates_by_diag_yr2)
##
## Deviance Residuals:
## [1] 0 0
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) 3.44848 0.04390 78.562 <2e-16 ***
## year8594Diagnosed 85-94 -0.12422 0.06476 -1.918 0.0551 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 3.6890e+00 on 1 degrees of freedom
## Residual deviance: 1.5543e-14 on 0 degrees of freedom
## AIC: 20.017
##
## Number of Fisher Scoring iterations: 2## exp(beta) 2.5 % 97.5 %
## (Intercept) 31.4525600 28.859880 34.278158
## year8594Diagnosed 85-94 0.8831851 0.777905 1.002714## Note that the scaling of the offset term only has an impact on the intercept
summary( glm( death_cancer ~ year8594 + offset( log( surv_mm ) ),
family=poisson, data=melanoma.l2 ) )##
## Call:
## glm(formula = death_cancer ~ year8594 + offset(log(surv_mm)),
## family = poisson, data = melanoma.l2)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.7931 -0.7568 -0.5632 -0.3081 3.0550
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -5.94418 0.04389 -135.426 <2e-16 ***
## year8594Diagnosed 85-94 -0.12422 0.06476 -1.918 0.0551 .
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 4814.3 on 5317 degrees of freedom
## Residual deviance: 4810.6 on 5316 degrees of freedom
## AIC: 6734.6
##
## Number of Fisher Scoring iterations: 6We see that Poisson regression is estimating the mortality rate ratio which, in this simple example, is the ratio of the two mortality rates.
The regression equation is:
\[\begin{align*} E(\text{death_cancer}) &= \frac{\text{surv_mm}}{12 \times 1000}\exp\left(\beta_0 + \beta_1 I(\text{year8594}="\text{Diagnosed 85-94}")\right) \\ &= \exp\left(\beta_0 + \beta_1 I(\text{year8594}="\text{Diagnosed 85-94}") + \log(\text{surv_mm}/1000/12) \right) \end{align*}\]
where \(E(\text{death_cancer})\) is the expected number of cancer deaths, \(\beta_0\) is the intercept term for the log rate, \(\beta_1\) is the log rate ratio for the later calendar period, and \(\text{surv_mm}/1000/12\) is the period-time.
(d)
In order to adjust for time since diagnosis (i.e., adjust for the fact that we expect mortality to depend on time since diagnosis) we need to split the data by this timescale. We will restrict our analysis to mortality up to 10 years following diagnosis.
## Add a new variable for year
melanoma.l2 <- mutate( melanoma.l2, surv_yy1 = surv_mm/12)
## Split follow up by year
melanoma.spl <- survSplit(melanoma.l2, cut=0:9, end="surv_yy1", start="start",
event="death_cancer")
## Calculate persontime and
## recode start time as a factor
melanoma.spl <- mutate(melanoma.spl,
pt = surv_yy1 - start,
fu = as.factor(start) )(e)
## Calculate the incidence rate by observation year
yearly_rates <- survRate(Surv(pt/1000,death_cancer)~fu, data=melanoma.spl)
## Plot by year
with(yearly_rates, matplot(fu,
cbind(rate, lower,
upper),
lty=c("solid","dashed","dashed"),
col=c("black","gray","gray"),
type="l",
main="Cancer deaths by years since diagnosis",
ylab="Incidence rate per 1000 person-years",
xlab="Years since diagnosis") )
It seems reasonable (at least to me) that melanoma-specific mortality is lower during the first year. These patients were classified as having localised skin melanoma at the time of diagnosis. That is, there was no evidence of metastases at the time of diagnosis although many of the patients who died would have had undetectable metastases or micrometastases at the time of diagnosis. It appears that it takes at least one year for these initially undetectable metastases to progress and cause the death of the patient.
(f)
# Plot smoothed hazards
par(mfrow=c(1,2))
with(yearly_rates, matplot(as.numeric(as.character(fu))+0.5,
cbind(rate, lower,
upper),
ylim=c(0,max(upper)),
lty=c("solid","dashed","dashed"),
col=c("black","gray","gray"),
type="l",
main="Cancer deaths by time since diagnosis",
ylab="Mortality rate per 1000 person-years",
xlab="Years since diagnosis") )
hazfit7f <- muhaz2(Surv(surv_mm/12, status == "Dead: cancer") ~ 1, data = melanoma.l)
## scale hazard by 1000
plot(hazfit7f, xlab="Years since diagnosis",col="blue",lty="solid", haz.scale=1000, xlim=c(0,10))
The pattern is similar. The plot of the mortality rates could be considered an approximation to the ‘true’ functional form depicted in the hazard plot. By estimating the rates for each year of follow-up we are essentially approximating the hazard using a step function. It would probably be more informative to use narrower intervals (e.g., 6-month intervals) for the first 6 months of follow-up.
(g)
## Run Poisson regression
summary(poisson7g <- glm( death_cancer ~ fu + offset( log(pt) ),
family = poisson,
data = melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu + offset(log(pt)), family = poisson,
## data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.3088 -0.2889 -0.2363 -0.1644 3.6805
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.3046 0.1187 -36.276 < 2e-16 ***
## fu1 1.2456 0.1359 9.166 < 2e-16 ***
## fu2 1.2617 0.1380 9.145 < 2e-16 ***
## fu3 1.0127 0.1460 6.934 4.08e-12 ***
## fu4 0.8186 0.1552 5.275 1.33e-07 ***
## fu5 0.7265 0.1630 4.456 8.36e-06 ***
## fu6 0.4961 0.1787 2.776 0.00551 **
## fu7 0.1682 0.2065 0.815 0.41531
## fu8 0.2886 0.2085 1.384 0.16641
## fu9 -0.2899 0.2765 -1.048 0.29452
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8446.4 on 34299 degrees of freedom
## AIC: 10386
##
## Number of Fisher Scoring iterations: 6## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.0135058 0.01070318 0.01704229
## fu1 3.4750768 2.66250895 4.53563117
## fu2 3.5312671 2.69465247 4.62762741
## fu3 2.7529574 2.06771439 3.66529071
## fu4 2.2673515 1.67274409 3.07332288
## fu5 2.0677380 1.50217318 2.84623672
## fu6 1.6422607 1.15697948 2.33108731
## fu7 1.1831887 0.78936945 1.77348567
## fu8 1.3345367 0.88679255 2.00834805
## fu9 0.7483554 0.43523195 1.28675268The pattern of the estimated mortality rate ratios mirrors the pattern we saw in the plot of the rates. Note that the first year of follow-up is the reference so the estimated rate ratio labelled 1 for fu is the rate ratio for the second year compared to the first year.
The regression equation is:
\[\begin{align*} E(\text{death_cancer}) &= \text{pt}\exp\left(\beta_0 + \beta_1 I(\text{fu}=1) + \beta_2 I(\text{fu}=2) + \beta_3 I(\text{fu}=3) + \beta_4 I(\text{fu}=4) + \beta_5 I(\text{fu}=5) + \beta_6 I(\text{fu}=6) + \beta_7 I(\text{fu}=7) + \beta_8 I(\text{fu}=8) + \beta_9 I(\text{fu}=9)\right) \\ &= \exp\left(\beta_0 + \beta_1 I(\text{fu}=1) + \beta_2 I(\text{fu}=2) + \beta_3 I(\text{fu}=3) + \beta_4 I(\text{fu}=4) + \beta_5 I(\text{fu}=5) + \beta_6 I(\text{fu}=6) + \beta_7 I(\text{fu}=7) + \beta_8 I(\text{fu}=8) + \beta_9 I(\text{fu}=9) + \log(\text{pt})\right) \end{align*}\]
(h)
summary(poisson7h <- glm( death_cancer ~ fu + year8594 + offset( log(pt) ),
family = poisson,
data = melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu + year8594 + offset(log(pt)),
## family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.3297 -0.2900 -0.2207 -0.1761 3.6707
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.16612 0.12380 -33.651 < 2e-16 ***
## fu1 1.24352 0.13589 9.151 < 2e-16 ***
## fu2 1.25370 0.13797 9.087 < 2e-16 ***
## fu3 0.99738 0.14610 6.827 8.68e-12 ***
## fu4 0.79438 0.15532 5.115 3.14e-07 ***
## fu5 0.69247 0.16329 4.241 2.23e-05 ***
## fu6 0.45104 0.17911 2.518 0.011796 *
## fu7 0.10844 0.20710 0.524 0.600559
## fu8 0.21049 0.20954 1.004 0.315139
## fu9 -0.39239 0.27780 -1.413 0.157802
## year8594Diagnosed 85-94 -0.24444 0.06579 -3.715 0.000203 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8432.6 on 34298 degrees of freedom
## AIC: 10375
##
## Number of Fisher Scoring iterations: 6## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01551228 0.01217011 0.01977229
## fu1 3.46780081 2.65693069 4.52614081
## fu2 3.50326901 2.67320128 4.59108479
## fu3 2.71116147 2.03608968 3.61005538
## fu4 2.21306295 1.63225462 3.00054142
## fu5 1.99864158 1.45125272 2.75249659
## fu6 1.56993640 1.10515372 2.23018778
## fu7 1.11453748 0.74268832 1.67256405
## fu8 1.23427730 0.81855780 1.86112752
## fu9 0.67543732 0.39185040 1.16425955
## year8594Diagnosed 85-94 0.78314061 0.68839639 0.89092450# Add interaction term
summary(poisson7h2 <- glm( death_cancer ~ fu*year8594 + offset( log(pt) ), family=poisson, data=melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu * year8594 + offset(log(pt)),
## family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.3546 -0.2682 -0.2244 -0.1764 3.6684
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.16333 0.17408 -23.917 < 2e-16 ***
## fu1 1.28028 0.19807 6.464 1.02e-10 ***
## fu2 1.39664 0.19729 7.079 1.45e-12 ***
## fu3 0.81978 0.21609 3.794 0.000148 ***
## fu4 0.65306 0.22519 2.900 0.003732 **
## fu5 0.66513 0.22711 2.929 0.003404 **
## fu6 0.41495 0.24264 1.710 0.087240 .
## fu7 0.03655 0.27163 0.135 0.892966
## fu8 0.30339 0.25453 1.192 0.233279
## fu9 -0.38675 0.31895 -1.213 0.225302
## year8594Diagnosed 85-94 -0.24965 0.23795 -1.049 0.294089
## fu1:year8594Diagnosed 85-94 -0.07110 0.27234 -0.261 0.794025
## fu2:year8594Diagnosed 85-94 -0.30852 0.27725 -1.113 0.265806
## fu3:year8594Diagnosed 85-94 0.34410 0.29326 1.173 0.240642
## fu4:year8594Diagnosed 85-94 0.29777 0.31086 0.958 0.338115
## fu5:year8594Diagnosed 85-94 0.06578 0.32894 0.200 0.841494
## fu6:year8594Diagnosed 85-94 0.09658 0.36443 0.265 0.790988
## fu7:year8594Diagnosed 85-94 0.22179 0.42829 0.518 0.604555
## fu8:year8594Diagnosed 85-94 -0.51530 0.53954 -0.955 0.339542
## fu9:year8594Diagnosed 85-94 -0.06040 0.79245 -0.076 0.939240
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8419.4 on 34289 degrees of freedom
## AIC: 10379
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01555564 0.01105892 0.0218808
## fu1 3.59765631 2.44018328 5.3041634
## fu2 4.04160662 2.74548881 5.9496087
## fu3 2.26998933 1.48622497 3.4670737
## fu4 1.92140436 1.23576375 2.9874599
## fu5 1.94474961 1.24608035 3.0351582
## fu6 1.51429324 0.94118270 2.4363857
## fu7 1.03722422 0.60906442 1.7663716
## fu8 1.35444106 0.82243832 2.2305753
## fu9 0.67926283 0.36353299 1.2692053
## year8594Diagnosed 85-94 0.77907144 0.48869327 1.2419903
## fu1:year8594Diagnosed 85-94 0.93136483 0.54614148 1.5883072
## fu2:year8594Diagnosed 85-94 0.73453606 0.42659693 1.2647612
## fu3:year8594Diagnosed 85-94 1.41072301 0.79400185 2.5064670
## fu4:year8594Diagnosed 85-94 1.34685580 0.73234442 2.4770047
## fu5:year8594Diagnosed 85-94 1.06799324 0.56050340 2.0349735
## fu6:year8594Diagnosed 85-94 1.10140218 0.53919080 2.2498284
## fu7:year8594Diagnosed 85-94 1.24831405 0.53921836 2.8899016
## fu8:year8594Diagnosed 85-94 0.59732398 0.20746996 1.7197475
## fu9:year8594Diagnosed 85-94 0.94138428 0.19917538 4.4493671The estimated mortality rate ratio is \(0.7791\) compared to \(0.8832\) (part c) and a value greater than 1 in part (b). The estimate we obtained in part (b) was subject to confounding by time-since-diagnosis. In part (c) we restricted to the first 10 years of follow-up subsequent to diagnosis. This did not, however, completely remove the confounding effect of time since diagnosis. There was still some confounding within the first 10 years of follow-up (if this is not clear to you then look in the data to see if there are associations between the confounder and the exposure and the confounder and the outcome) so the estimate was subject to residual confounding. Now, when we adjust for time since diagnosis we see that the estimate changes further.
(i)
Now control for age, sex, and calendar period. Write out the regression equation.
The regression equation is: \[\begin{align*} E(\text{death_cancer}) &= \text{pt}\exp\left(\beta_0 + \beta_1 I(\text{fu}=1) + \beta_2 I(\text{fu}=2) + \beta_3 I(\text{fu}=3) + \beta_4 I(\text{fu}=4) + \beta_5 I(\text{fu}=5) + \beta_6 I(\text{fu}=6) + \beta_7 I(\text{fu}=7) + \beta_8 I(\text{fu}=8) + \beta_9 I(\text{fu}=9) + \right. \\ &\qquad \left.\beta_{10} x + \beta_{11} I(\text{fu}=1) x + \beta_{12} I(\text{fu}=2) x + \beta_{13} I(\text{fu}=3) x + \beta_{14} I(\text{fu}=4) x + \beta_{15} I(\text{fu}=5) x + \beta_{16} I(\text{fu}=6) x + \beta_{17} I(\text{fu}=7) x + \beta_{18} I(\text{fu}=8) x + \beta_{19} I(\text{fu}=9) x \right) \end{align*}\] where \(x\) is the indicator variable when year8594=“Diagnosed 85-94”.
i.
For patients of the same sex diagnosed in the same calendar period, those aged 60–74 at diagnosis have an estimated 86% higher risk of death due to skin melanoma than those aged 0–44 at diagnosis. The difference is statistically significant.
ii.
The parameter estimate for period changes from 0.78 to 0.72 when age and sex are added to the model. Whether this is ‘strong confounding’, or even ‘confounding’ is a matter of judgement. I would consider this confounding but not strong confounding but there is no correct answer.
iii.
##
## Call:
## glm(formula = death_cancer ~ fu + year8594 + sex + agegrp + offset(log(pt)),
## family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5646 -0.2672 -0.2098 -0.1597 3.6806
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.36681 0.14322 -30.490 < 2e-16 ***
## fu1 1.26827 0.13592 9.331 < 2e-16 ***
## fu2 1.30657 0.13806 9.464 < 2e-16 ***
## fu3 1.07575 0.14627 7.354 1.92e-13 ***
## fu4 0.89517 0.15559 5.753 8.75e-09 ***
## fu5 0.81370 0.16368 4.971 6.65e-07 ***
## fu6 0.58637 0.17957 3.265 0.00109 **
## fu7 0.25361 0.20758 1.222 0.22181
## fu8 0.36427 0.21006 1.734 0.08290 .
## fu9 -0.22796 0.27844 -0.819 0.41296
## year8594Diagnosed 85-94 -0.32516 0.06618 -4.913 8.97e-07 ***
## sexFemale -0.53180 0.06545 -8.125 4.48e-16 ***
## agegrp45-59 0.28352 0.09417 3.011 0.00261 **
## agegrp60-74 0.62185 0.09088 6.843 7.76e-12 ***
## agegrp75+ 1.22386 0.10444 11.718 < 2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8233.4 on 34294 degrees of freedom
## AIC: 10183
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01269168 0.009585413 0.01680457
## fu1 3.55468470 2.723340923 4.63980959
## fu2 3.69349752 2.817870250 4.84121792
## fu3 2.93219656 2.201336558 3.90570748
## fu4 2.44775331 1.804376456 3.32053559
## fu5 2.25623262 1.637030417 3.10964634
## fu6 1.79745329 1.264170006 2.55569926
## fu7 1.28866663 0.857919555 1.93568462
## fu8 1.43945962 0.953660962 2.17272602
## fu9 0.79615726 0.461304916 1.37407245
## year8594Diagnosed 85-94 0.72241051 0.634523266 0.82247095
## sexFemale 0.58754651 0.516807578 0.66796796
## agegrp45-59 1.32779475 1.104004888 1.59694845
## agegrp60-74 1.86237635 1.558526802 2.22546423
## agegrp75+ 3.40028687 2.770846371 4.17271449## Test if the effect of age is significant using a likelihood ratio test
drop1(poisson7i, ~agegrp, test="Chisq")## Single term deletions
##
## Model:
## death_cancer ~ fu + year8594 + sex + agegrp + offset(log(pt))
## Df Deviance AIC LRT Pr(>Chi)
## <none> 8233.4 10183
## agegrp 3 8377.9 10322 144.59 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## For this we can also use the car package and a Wald test
linearHypothesis(poisson7i,c("agegrp45-59 = 0","agegrp60-74 = 0","agegrp75+ = 0"))## Linear hypothesis test
##
## Hypothesis:
## agegrp45 - 59 = 0
## agegrp60 - 74 = 0
## agegrp75 + = 0
##
## Model 1: restricted model
## Model 2: death_cancer ~ fu + year8594 + sex + agegrp + offset(log(pt))
##
## Res.Df Df Chisq Pr(>Chisq)
## 1 34297
## 2 34294 3 155.82 < 2.2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1## ADVANCED:
## Alternative approach for the likelihood ratio test
# poisson7i_2 <- update(poisson7i,. ~ . - agegrp)
# anova(poisson7i_2,poisson7i,test="Chisq")Age (modelled as a categorical variable with 4 levels) is highly significant in the model.
(j)
##
## Call:
## glm(formula = death_cancer ~ fu + agegrp + year8594 * sex + offset(log(pt)),
## family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5611 -0.2689 -0.2108 -0.1593 3.6840
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.37900 0.14596 -30.001 < 2e-16 ***
## fu1 1.26830 0.13592 9.331 < 2e-16 ***
## fu2 1.30659 0.13806 9.464 < 2e-16 ***
## fu3 1.07569 0.14627 7.354 1.92e-13 ***
## fu4 0.89511 0.15559 5.753 8.77e-09 ***
## fu5 0.81360 0.16369 4.971 6.68e-07 ***
## fu6 0.58630 0.17958 3.265 0.001095 **
## fu7 0.25340 0.20759 1.221 0.222197
## fu8 0.36405 0.21007 1.733 0.083090 .
## fu9 -0.22829 0.27845 -0.820 0.412297
## agegrp45-59 0.28270 0.09419 3.001 0.002688 **
## agegrp60-74 0.62118 0.09089 6.835 8.23e-12 ***
## agegrp75+ 1.22364 0.10444 11.717 < 2e-16 ***
## year8594Diagnosed 85-94 -0.29917 0.08840 -3.384 0.000714 ***
## sexFemale -0.50562 0.08813 -5.737 9.64e-09 ***
## year8594Diagnosed 85-94:sexFemale -0.05792 0.13061 -0.443 0.657440
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8233.2 on 34293 degrees of freedom
## AIC: 10185
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01253791 0.009418543 0.01669039
## fu1 3.55479530 2.723425214 4.63995469
## fu2 3.69354707 2.817905687 4.84128693
## fu3 2.93201254 2.201195141 3.90546816
## fu4 2.44760423 1.804262241 3.32034133
## fu5 2.25601958 1.636868208 3.10936723
## fu6 1.79732528 1.264071466 2.55553445
## fu7 1.28840069 0.857735550 1.93530086
## fu8 1.43915154 0.953447823 2.17228159
## fu9 0.79589580 0.461149435 1.37363308
## agegrp45-59 1.32670920 1.103059120 1.59570531
## agegrp60-74 1.86113111 1.557443264 2.22403543
## agegrp75+ 3.39953913 2.770276830 4.17173697
## year8594Diagnosed 85-94 0.74143513 0.623488755 0.88169360
## sexFemale 0.60313385 0.507452579 0.71685602
## year8594Diagnosed 85-94:sexFemale 0.94372451 0.730577239 1.21905789The interaction term is not statistically significant indicating that there is no evidence that the effect of sex is modified by period.
(k)
Based on the interaction model you fitted in exercise 7j, estimate the hazard ratio for the effect of sex (with 95% confidence interval) for each calendar period.
ADVANCED: Do this with each of the following methods and confirm that the results are the same:
i.
## sexFemale
## 0.6031338## sexFemale
## 0.5691922The effect of sex for patients diagnosed 1975–84 is \(0.6031338\) and the effect of sex for patients diagnosed 1985–94 is \(0.6031338 \times 0.9437245=0.56919214\).
ii.
We can use lincom to get the estimated effect for patients diagnosed 1985–94.
## You will need the "car" package to use lincom. If it is not already installed:
## install.packages("car")
lincom(poisson7j,c("sexFemale + year8594Diagnosed 85-94:sexFemale"),eform=TRUE)## Estimate 2.5 % 97.5 %
## sexFemale + year8594Diagnosed 85-94:sexFemale 0.5691922 0.4705541 0.6885069
## Chisq Pr(>Chisq)
## sexFemale + year8594Diagnosed 85-94:sexFemale 33.68456 6.481293e-09The advantage of lincom is that we also get a confidence interval (not easy to calculate by hand since the SE is a function of variances and covariances).
iii.
## Create dummies and Poisson regression
melanoma.spl <- melanoma.spl %>%
## Add confidence intervals for the rates
mutate(femaleEarly = sex=="Female" & year8594=="Diagnosed 75-84",
femaleLate = sex=="Female" & year8594=="Diagnosed 85-94")
summary(poisson7k <- glm( death_cancer ~ fu + agegrp + year8594 + femaleEarly +
femaleLate + offset( log(pt) ), family=poisson,
data=melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu + agegrp + year8594 + femaleEarly +
## femaleLate + offset(log(pt)), family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5611 -0.2689 -0.2108 -0.1593 3.6840
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.37900 0.14596 -30.001 < 2e-16 ***
## fu1 1.26830 0.13592 9.331 < 2e-16 ***
## fu2 1.30659 0.13806 9.464 < 2e-16 ***
## fu3 1.07569 0.14627 7.354 1.92e-13 ***
## fu4 0.89511 0.15559 5.753 8.77e-09 ***
## fu5 0.81360 0.16369 4.971 6.68e-07 ***
## fu6 0.58630 0.17958 3.265 0.001095 **
## fu7 0.25340 0.20759 1.221 0.222197
## fu8 0.36405 0.21007 1.733 0.083090 .
## fu9 -0.22829 0.27845 -0.820 0.412297
## agegrp45-59 0.28270 0.09419 3.001 0.002688 **
## agegrp60-74 0.62118 0.09089 6.835 8.23e-12 ***
## agegrp75+ 1.22364 0.10444 11.717 < 2e-16 ***
## year8594Diagnosed 85-94 -0.29917 0.08840 -3.384 0.000714 ***
## femaleEarlyTRUE -0.50562 0.08813 -5.737 9.64e-09 ***
## femaleLateTRUE -0.56354 0.09710 -5.804 6.48e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8233.2 on 34293 degrees of freedom
## AIC: 10185
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01253791 0.009418543 0.01669039
## fu1 3.55479530 2.723425214 4.63995469
## fu2 3.69354707 2.817905687 4.84128693
## fu3 2.93201254 2.201195141 3.90546816
## fu4 2.44760423 1.804262241 3.32034133
## fu5 2.25601958 1.636868208 3.10936723
## fu6 1.79732528 1.264071466 2.55553445
## fu7 1.28840069 0.857735550 1.93530086
## fu8 1.43915154 0.953447823 2.17228159
## fu9 0.79589580 0.461149435 1.37363308
## agegrp45-59 1.32670920 1.103059120 1.59570531
## agegrp60-74 1.86113111 1.557443264 2.22403543
## agegrp75+ 3.39953913 2.770276830 4.17173697
## year8594Diagnosed 85-94 0.74143513 0.623488755 0.88169360
## femaleEarlyTRUE 0.60313385 0.507452579 0.71685602
## femaleLateTRUE 0.56919219 0.470554120 0.68850689iv.
## Add interaction term
summary(poisson7k2 <- glm( death_cancer ~ fu + agegrp + year8594 + year8594:sex +
offset( log(pt) ), family=poisson,
data=melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu + agegrp + year8594 + year8594:sex +
## offset(log(pt)), family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5611 -0.2689 -0.2108 -0.1593 3.6840
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.37900 0.14596 -30.001 < 2e-16 ***
## fu1 1.26830 0.13592 9.331 < 2e-16 ***
## fu2 1.30659 0.13806 9.464 < 2e-16 ***
## fu3 1.07569 0.14627 7.354 1.92e-13 ***
## fu4 0.89511 0.15559 5.753 8.77e-09 ***
## fu5 0.81360 0.16369 4.971 6.68e-07 ***
## fu6 0.58630 0.17958 3.265 0.001095 **
## fu7 0.25340 0.20759 1.221 0.222197
## fu8 0.36405 0.21007 1.733 0.083090 .
## fu9 -0.22829 0.27845 -0.820 0.412297
## agegrp45-59 0.28270 0.09419 3.001 0.002688 **
## agegrp60-74 0.62118 0.09089 6.835 8.23e-12 ***
## agegrp75+ 1.22364 0.10444 11.717 < 2e-16 ***
## year8594Diagnosed 85-94 -0.29917 0.08840 -3.384 0.000714 ***
## year8594Diagnosed 75-84:sexFemale -0.50562 0.08813 -5.737 9.64e-09 ***
## year8594Diagnosed 85-94:sexFemale -0.56354 0.09710 -5.804 6.48e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8233.2 on 34293 degrees of freedom
## AIC: 10185
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01253791 0.009418543 0.01669039
## fu1 3.55479530 2.723425214 4.63995469
## fu2 3.69354707 2.817905687 4.84128693
## fu3 2.93201254 2.201195141 3.90546816
## fu4 2.44760423 1.804262241 3.32034133
## fu5 2.25601958 1.636868208 3.10936723
## fu6 1.79732528 1.264071466 2.55553445
## fu7 1.28840069 0.857735550 1.93530086
## fu8 1.43915154 0.953447823 2.17228159
## fu9 0.79589580 0.461149435 1.37363308
## agegrp45-59 1.32670920 1.103059120 1.59570531
## agegrp60-74 1.86113111 1.557443264 2.22403543
## agegrp75+ 3.39953913 2.770276830 4.17173697
## year8594Diagnosed 85-94 0.74143513 0.623488755 0.88169360
## year8594Diagnosed 75-84:sexFemale 0.60313385 0.507452579 0.71685602
## year8594Diagnosed 85-94:sexFemale 0.56919219 0.470554120 0.68850689(l)
If we fit stratified models we get slightly different estimates (\(0.6165815\) and \(0.5549737\)) since the models stratified by calendar period imply that all estimates are modified by calendar period. That is, we are actually estimating the following model:
summary( poisson7l.early <- glm( death_cancer ~ fu + agegrp + sex + offset( log(pt) ),
family = poisson, data = melanoma.spl,
subset = year8594 == "Diagnosed 75-84" ) )##
## Call:
## glm(formula = death_cancer ~ fu + agegrp + sex + offset(log(pt)),
## family = poisson, data = melanoma.spl, subset = year8594 ==
## "Diagnosed 75-84")
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5488 -0.2801 -0.2186 -0.1698 3.6866
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.35024 0.19749 -22.028 < 2e-16 ***
## fu1 1.29711 0.19807 6.549 5.80e-11 ***
## fu2 1.43389 0.19734 7.266 3.70e-13 ***
## fu3 0.87511 0.21622 4.047 5.18e-05 ***
## fu4 0.72162 0.22538 3.202 0.00137 **
## fu5 0.74575 0.22738 3.280 0.00104 **
## fu6 0.50579 0.24297 2.082 0.03737 *
## fu7 0.13806 0.27198 0.508 0.61172
## fu8 0.41333 0.25503 1.621 0.10508
## fu9 -0.26674 0.31931 -0.835 0.40352
## agegrp45-59 0.36623 0.12112 3.024 0.00250 **
## agegrp60-74 0.59417 0.11933 4.979 6.39e-07 ***
## agegrp75+ 1.02300 0.15322 6.677 2.45e-11 ***
## sexFemale -0.48356 0.08839 -5.471 4.48e-08 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 4649.9 on 16933 degrees of freedom
## Residual deviance: 4430.6 on 16920 degrees of freedom
## AIC: 5496.6
##
## Number of Fisher Scoring iterations: 6## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01290368 0.008762127 0.01900279
## fu1 3.65871653 2.481590140 5.39420529
## fu2 4.19499842 2.849396206 6.17604941
## fu3 2.39915019 1.570414849 3.66522365
## fu4 2.05775938 1.322964109 3.20067161
## fu5 2.10802429 1.349983775 3.29171839
## fu6 1.65830219 1.030014324 2.66983293
## fu7 1.14804722 0.673675578 1.95644975
## fu8 1.51184459 0.917120868 2.49222775
## fu9 0.76587332 0.409596432 1.43204847
## agegrp45-59 1.44229093 1.137518383 1.82872044
## agegrp60-74 1.81153335 1.433732925 2.28888730
## agegrp75+ 2.78152101 2.059958553 3.75583242
## sexFemale 0.61658148 0.518507340 0.73320605summary( poisson7l.late <- glm( death_cancer ~ fu + agegrp + sex + offset( log(pt) ),
family = poisson, data = melanoma.spl,
subset = year8594 == "Diagnosed 85-94" ) )##
## Call:
## glm(formula = death_cancer ~ fu + agegrp + sex + offset(log(pt)),
## family = poisson, data = melanoma.spl, subset = year8594 ==
## "Diagnosed 85-94")
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5041 -0.2578 -0.1938 -0.1457 3.6334
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.716255 0.198840 -23.719 < 2e-16 ***
## fu1 1.242842 0.186946 6.648 2.97e-11 ***
## fu2 1.158910 0.194930 5.945 2.76e-09 ***
## fu3 1.269035 0.198560 6.391 1.65e-10 ***
## fu4 1.091821 0.214814 5.083 3.72e-07 ***
## fu5 0.907839 0.238726 3.803 0.000143 ***
## fu6 0.712226 0.272810 2.611 0.009036 **
## fu7 0.467548 0.331976 1.408 0.159019
## fu8 0.003177 0.476361 0.007 0.994678
## fu9 -0.219501 0.725878 -0.302 0.762352
## agegrp45-59 0.170696 0.149540 1.141 0.253672
## agegrp60-74 0.657696 0.140647 4.676 2.92e-06 ***
## agegrp75+ 1.384288 0.148765 9.305 < 2e-16 ***
## sexFemale -0.588835 0.097576 -6.035 1.59e-09 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 3997.8 on 17374 degrees of freedom
## Residual deviance: 3780.7 on 17361 degrees of freedom
## AIC: 4690.7
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.008948632 0.006060435 0.01321324
## fu1 3.465448196 2.402316723 4.99906240
## fu2 3.186458106 2.174619253 4.66910023
## fu3 3.557416952 2.410572145 5.24988037
## fu4 2.979696169 1.955790301 4.53964275
## fu5 2.478958882 1.552620335 3.95797801
## fu6 2.038523435 1.194262121 3.47961953
## fu7 1.596076214 0.832673832 3.05937233
## fu8 1.003182304 0.394367397 2.55187103
## fu9 0.802919497 0.193554438 3.33074109
## agegrp45-59 1.186130380 0.884797127 1.59008798
## agegrp60-74 1.930340028 1.465259625 2.54303917
## agegrp75+ 3.991981639 2.982359282 5.34339290
## sexFemale 0.554973666 0.458369470 0.67193779## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01269168 0.009585413 0.01680457
## fu1 3.55468470 2.723340923 4.63980959
## fu2 3.69349752 2.817870250 4.84121792
## fu3 2.93219656 2.201336558 3.90570748
## fu4 2.44775331 1.804376456 3.32053559
## fu5 2.25623262 1.637030417 3.10964634
## fu6 1.79745329 1.264170006 2.55569926
## fu7 1.28866663 0.857919555 1.93568462
## fu8 1.43945962 0.953660962 2.17272602
## fu9 0.79615726 0.461304916 1.37407245
## year8594Diagnosed 85-94 0.72241051 0.634523266 0.82247095
## sexFemale 0.58754651 0.516807578 0.66796796
## agegrp45-59 1.32779475 1.104004888 1.59694845
## agegrp60-74 1.86237635 1.558526802 2.22546423
## agegrp75+ 3.40028687 2.770846371 4.17271449## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01253791 0.009418543 0.01669039
## fu1 3.55479530 2.723425214 4.63995469
## fu2 3.69354707 2.817905687 4.84128693
## fu3 2.93201254 2.201195141 3.90546816
## fu4 2.44760423 1.804262241 3.32034133
## fu5 2.25601958 1.636868208 3.10936723
## fu6 1.79732528 1.264071466 2.55553445
## fu7 1.28840069 0.857735550 1.93530086
## fu8 1.43915154 0.953447823 2.17228159
## fu9 0.79589580 0.461149435 1.37363308
## agegrp45-59 1.32670920 1.103059120 1.59570531
## agegrp60-74 1.86113111 1.557443264 2.22403543
## agegrp75+ 3.39953913 2.770276830 4.17173697
## year8594Diagnosed 85-94 0.74143513 0.623488755 0.88169360
## sexFemale 0.60313385 0.507452579 0.71685602
## year8594Diagnosed 85-94:sexFemale 0.94372451 0.730577239 1.21905789# Poisson-regression with effects specific for diagnose period
summary(poisson7l2 <- glm( death_cancer ~ fu + fu:year8594 + agegrp + agegrp:year8594
+ sex*year8594 + offset( log(pt) ),
family=poisson, data=melanoma.spl ))##
## Call:
## glm(formula = death_cancer ~ fu + fu:year8594 + agegrp + agegrp:year8594 +
## sex * year8594 + offset(log(pt)), family = poisson, data = melanoma.spl)
##
## Deviance Residuals:
## Min 1Q Median 3Q Max
## -0.5488 -0.2630 -0.2067 -0.1606 3.6866
##
## Coefficients:
## Estimate Std. Error z value Pr(>|z|)
## (Intercept) -4.35024 0.19751 -22.026 < 2e-16 ***
## fu1 1.29711 0.19808 6.548 5.82e-11 ***
## fu2 1.43389 0.19736 7.266 3.72e-13 ***
## fu3 0.87511 0.21623 4.047 5.18e-05 ***
## fu4 0.72162 0.22540 3.202 0.00137 **
## fu5 0.74575 0.22740 3.280 0.00104 **
## fu6 0.50579 0.24299 2.082 0.03738 *
## fu7 0.13806 0.27202 0.508 0.61177
## fu8 0.41333 0.25504 1.621 0.10510
## fu9 -0.26674 0.31945 -0.835 0.40372
## agegrp45-59 0.36623 0.12113 3.024 0.00250 **
## agegrp60-74 0.59417 0.11934 4.979 6.40e-07 ***
## agegrp75+ 1.02300 0.15323 6.676 2.45e-11 ***
## sexFemale -0.48356 0.08839 -5.471 4.48e-08 ***
## year8594Diagnosed 85-94 -0.36601 0.28026 -1.306 0.19156
## fu1:year8594Diagnosed 85-94 -0.05427 0.27237 -0.199 0.84207
## fu2:year8594Diagnosed 85-94 -0.27498 0.27739 -0.991 0.32153
## fu3:year8594Diagnosed 85-94 0.39392 0.29357 1.342 0.17965
## fu4:year8594Diagnosed 85-94 0.37020 0.31137 1.189 0.23445
## fu5:year8594Diagnosed 85-94 0.16209 0.32970 0.492 0.62298
## fu6:year8594Diagnosed 85-94 0.20643 0.36533 0.565 0.57204
## fu7:year8594Diagnosed 85-94 0.32949 0.42919 0.768 0.44267
## fu8:year8594Diagnosed 85-94 -0.41015 0.54034 -0.759 0.44781
## fu9:year8594Diagnosed 85-94 0.04724 0.79306 0.060 0.95250
## year8594Diagnosed 85-94:agegrp45-59 -0.19554 0.19244 -1.016 0.30959
## year8594Diagnosed 85-94:agegrp60-74 0.06352 0.18446 0.344 0.73056
## year8594Diagnosed 85-94:agegrp75+ 0.36129 0.21357 1.692 0.09070 .
## year8594Diagnosed 85-94:sexFemale -0.10527 0.13166 -0.800 0.42397
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
##
## (Dispersion parameter for poisson family taken to be 1)
##
## Null deviance: 8651.5 on 34308 degrees of freedom
## Residual deviance: 8211.4 on 34281 degrees of freedom
## AIC: 10187
##
## Number of Fisher Scoring iterations: 7## exp(beta) 2.5 % 97.5 %
## (Intercept) 0.01290368 0.008761852 0.01900338
## fu1 3.65871657 2.481529705 5.39433679
## fu2 4.19499847 2.849324333 6.17620534
## fu3 2.39915022 1.570376752 3.66531265
## fu4 2.05775940 1.322929450 3.20075553
## fu5 2.10802431 1.349942800 3.29181835
## fu6 1.65830221 1.029983069 2.66991399
## fu7 1.14804717 0.673627936 1.95658794
## fu8 1.51184461 0.917092926 2.49230372
## fu9 0.76587284 0.409489206 1.43242168
## agegrp45-59 1.44229095 1.137497545 1.82875400
## agegrp60-74 1.81153338 1.433707489 2.28892799
## agegrp75+ 2.78152106 2.059922700 3.75589792
## sexFemale 0.61658148 0.518502522 0.73321286
## year8594Diagnosed 85-94 0.69349480 0.400391639 1.20116153
## fu1:year8594Diagnosed 85-94 0.94717591 0.555376681 1.61537607
## fu2:year8594Diagnosed 85-94 0.75958505 0.441020116 1.30826106
## fu3:year8594Diagnosed 85-94 1.48278208 0.834053707 2.63609247
## fu4:year8594Diagnosed 85-94 1.44802943 0.786579357 2.66570589
## fu5:year8594Diagnosed 85-94 1.17596314 0.616248847 2.24404364
## fu6:year8594Diagnosed 85-94 1.22928344 0.600729103 2.51550617
## fu7:year8594Diagnosed 85-94 1.39025317 0.599473921 3.22416674
## fu8:year8594Diagnosed 85-94 0.66354855 0.230109662 1.91342111
## fu9:year8594Diagnosed 85-94 1.04837181 0.221544163 4.96101286
## year8594Diagnosed 85-94:agegrp45-59 0.82239328 0.563992308 1.19918426
## year8594Diagnosed 85-94:agegrp60-74 1.06558347 0.742295370 1.52967158
## year8594Diagnosed 85-94:agegrp75+ 1.43517937 0.944316163 2.18119726
## year8594Diagnosed 85-94:sexFemale 0.90008164 0.695365474 1.16506642