Using the Results Given by Lu and Zeger,14 the Expectation of the Estimating Equation For

Appendix 1

Using the results given by Lu and Zeger,14 the expectation of the estimating equation for the case-crossover design can be written as

  T  exp(X  S ) EU ( )  X e X t exp(S )  m m  t  t  exp(X )  t1  m(t)  j   jW (m)  T X t ˆ   X t e exp(St )  E[exp(St ( ))] t1 T X t St   X t e e ( ) t1

St where e ( ) is the difference between true exp(St) and the expectation of the estimated

ˆ exp(St ( )) , W(m) is the referent window for day, and (t) is the set of days containing day t in their referent window.

1 Appendix 2

We model the daily average PM10 in Chicago from January 1st, 1995 to December

31st, 1996 as the exposure Xt. We simulate Yt, daily mortality, from a Poisson t, where

logt=Xt+St. We set  equal to a change of 0, 1, 2, and 5 percent in daily mortality per

3 10 g/m increase in PM10 for three functions St described below. Data and software for this simulation are available at the iHAPPS website.

Scenario I: St is constant (St =4.10) over time with value equal to the natural logarithm of mean daily mortality for people 75 years and older in Chicago for the two year study period.

Scenario II: The baseline risk increases over time, and there exists a seasonal trend that peaks in winter. St is a combination of a linear function of time and a cosine function

with period one year (St=4.10+0.0001(t-365)+0.1cos(2t/365)).

Scenario III: Scenario II plus additional DOW effects, Sdow, t, for Sunday through

Saturday that equal 0.000, 0.002, 0.010, 0.008, 0.005, 0.008, and -0.005, respectively.

These values are close to those estimated from the Chicago data.

Four models were estimated for each scenario.

Model 1: a time-stratified case-crossover design (TSD) using only those days with the same day of the week in the same calendar month and year of the event day as control days.

2 Model 2: a time series with logt=Xt+dowtns(t,df=12 per year), where ns is natural spline, and dowt is the 7 level indicator for DOW. Model 2 has similar degrees of freedom as model 1 but allows the DOW effect to vary as a smooth function over time.

Model 3: a time series with logt=Xt+dowtns(t, df = 4 per year). Model 3 has the same form as model 2 with fewer degrees of freedom than the other three models.

We used 10000 replicates for each scenario. The estimated standard deviation and bias for each model as well as the ratio of mean squared error comparing with Model

3 are reported in Table 1. The standard deviations for the estimates are similar across the

3 assumed scenarios and 4 values of  for the same model. The order of the magnitude for the standard deviations for ˆ is: Model 3 < Model 1< Model 2.

The biases for ˆ vary across the scenarios for each  value, but the magnitude of the bias is generally much smaller than the standard deviation. The mean squared errors from the simulation suggested that using too many degrees of freedom could make the time series estimator worse than case-crossover method due to the bias variance trade-off.

The bias listed in the table is the deviation of the expected regression coefficient from its true  value. For scenario I, the three models all have very small estimates of bias.

When the true St is constant, the TSD assumption that St is a step function is correct, hence the TSD estimators are unbiased. For scenarios II and III, the time series models have much smaller bias than the TSD model. When the true St has a trend, such as seasonal or DOW effects, the TSD case-crossover model cannot capture the smoothness

of St, producing bias in the estimate of ˆ .

3 In summary, whenever a model, be it time series or case-crossover, fails to appropriately account for fluctuations in time that confound the exposure, bias in the effect estimate is produced. It is therefore important to perform model-checking to TSD case-crossover analyses rather than assuming that the estimator is unbiased.

4 TABLE 1. Bias and standard deviation of the estimate and the ratio of mean squared errors comparing to Model 3 (time series model with 4df/year) from the simulation study (10000 simulations).

Model 1 Model 2 Model 3

(TSD) (TS 12df/year) (TS 4df/year)

Scenario  Bias SD MSE Bias SD MSE Bias SD MSE

ratio ratio ratio

Scenario I 0 4 346 1.25 -2.2 351 1.28 2.7 309 1

1 -0.2 347 1.27 1.2 352 1.31 -0.4 308 1 Constant St 2 0.2 342 1.25 5e-3 347 1.29 -0.8 306 1

5 -0.2 332 1.28 3e-2 337 1.32 -1.4 294 1

Scenario II 0 -31 351 1.25 -0.4 358 1.29 -2.5 314 1

1 -34 349 1.27 -2.9 353 1.28 -6.0 312 1 Trend + 2 -31 346 1.26 0.6 352 1.29 -2.4 309 1

seasonality 5 -28 338 1.29 2.8 342 1.31 -1.7 299 1

Scenario III 0 -26 351 1.24 4.5 356 1.27 3.9 317 1

1 -26 346 1.26 5.2 351 1.29 3.6 309 1 Trend + 2 -26 342 1.27 5.0 347 1.30 3.8 304 1

Seasonality + 5 -27 333 1.26 3.9 338 1.29 0.3 298 1

day-of-week