Multiple membership analysis using neonatal data

 1

M − 1

M m=1

( ˆβ_m−β)¯ˆ ² (5.5)

and the mean square error is given by

(β¯ˆ− β)²+ SE( ˆβ)² (5.6)

The coverage is percentage of the M datasets where the 95% credible interval includes the known true parameter.

To examine the impact on provider effects we present the bias, MSE and coverage averaged over all 100 providers. The average posterior SD over all providers is presented as a measure of the uncertainty around provider effects. To investigate whether the Deviance Information Criterion (DIC) (Spiegelhalter et al., 2002) discriminates between the different models, I present the mean DIC and the proportion of datasets where the correct analysis model is chosen based on DIC (applying a minimum difference of 5 (Spiegelhalter et al., 2003)).

5.4 Multiple membership analysis using neonatal data

5.4.1 Aim

The aim of this section is to describe methods to estimate NNU effects on mortality for singleton preterm infants, including those who were transferred. I applied MM models described in section 4.3, building on the methods for the hierarchical analysis of non-transferred infants described in section 5.2. I explored sensitivity to different weights, as was done with simulated data in section 5.3.

5.4.2 Weighting schemes

In addition to the four analysis weightings used in the simulation study (LOS, Beta2, Beta3,First ) I applied two other weightings which are relevant for provider profiling purposes. As explained in chapter 1, for each day an infant is in the neonatal unit their care is classified as intensive care (IC), high dependency care (HDC) or special care

(SC) in decreasing order of intensity. Therefore we can augment the LOS weighting described in section 5.3 by assigning weight according to the intensity of care received each day. This weighted length of stay approach is labelled WLOS for convenience. For each infant, each day in neonatal care (across all NNUs providing care) was assigned a weight of 1 for IC, 0.5 for HDC and 0.25 for SC. These ratios are the nurse staffing ratios recommended by the British Association of Perinatal Medicine (1:1 nurse to infant ratio for IC, 1:2 for HDC and 1:4 for SC, see section 1.2). Letting d^IC_ij denote the number of IC days infant i has in NNU j, and d^IC_ij the total number of IC days for infant i (with similar definitions for HDC and SC) then the weights are then given by:

w_ij = d^IC_ij + 0.5d^HDC_ij + 0.25d^SC_ij d^IC_i + 0.5d^HDC_i + 0.25d^SC_i

As discussed in chapter 4, in some provider profiling analyses which did not use hierarchical models, individuals were assigned to all providers that treated them. A similar approach for multiple membership models is to assign individuals to all providers equally regardless of length of stay, i.e.

w_ij = 1/N_i ∀j ∈ U(i)

where U (i) denotes the set of N_i providers treating individual i. This weighting is given the label Equal. A summary of all six weights is shown in table 5.1.

Table 5.1: Weightings for multiple membership models for simulated and neonatal data

Label Description

LOS The outcome is attributed in proportion to the length of stay

Beta2 The outcome is attributed using the Beta(1,2) weight Beta3 The outcome is attributed using the Beta(1,3) weight First The outcome is attributed to the first provider only WLOS

(neonatal data only) The outcome is attributed according to length of stay weighted by the type of care received, using recommended nurse staffing ratios

Equal

(neonatal data only) The outcome is attributed equally to all NNUs caring for the infant, regardless of length of stay or intensity of care

5.4.3 Data

The data used in this chapter are a subset of those described in 2.3.1 restricted to singleton infants with no missing data, as for the analysis of non-transferred infants described in 5.2. I also excluded any infants without a complete episodic record with all

care in English NNUs. Episodic records were determined as incomplete if there was a missing episode of care based on the episode number, or by discrepant details (more than 24 hours between discharge from one NNU and admission to another, or if the discharge destination NNU was different to the NNU of the next episodic record). Infants who had an episode where the episode number, admission or discharge times, the NNU providing care or daily care level data were missing were also excluded. A brief description of this cohort is provided in chapter 7. I summarised the distribution of the total weight for all infants across NNU levels and episodes of care to illustrate how the weighting schemes differ in practice.

5.4.4 Statistical models and estimation

As for the analysis of non-transferred infants, risk factors included in the model were those determined in chapter 3: gestational age at birth, birth weight, sex, and whether the mother received antenatal steroids, with spline terms and interactions as described in 5.2. Modelling strategies for neonatal network effects, variation by level of NNU, and prior distributions for the random effects were guided by the findings of the analysis described in 5.2. Using Model 1 (equation 5.2) as an example we have:

logit(p_i) = β₁g_i⁽¹⁾+ β₂g_i⁽²⁾+ β₃b_i+ β₄b²_i + β₅g_i⁽¹⁾b_i+ β₆g_i⁽¹⁾b²_i + β₇b_ig_i⁽²⁾ + β₈m_i+ β₉s_i+

j∈U(i)w_ijμ_j i = 1, ..., N

μ_j = α + u_j j = 1, ..., J

Priors

α∼ N(0, 100000)

β_s∼ N(0, 100000) s = 1, ..., 9

u_j ∼ N(0, σ_u²) Hyperprior

σ_u ∼ U(0, 5)

Note that, as described when these models were introduced in 4.3, we subscript any quantities at the infant level with i rather than ij as the infant is no longer nested in a single NNU. Multiple membership for the other models follows in the same way (the μ_j is replaced by 

j∈U(i)w_ijμ_j).

Six different models were run, one for each of the weighting schemes described in table 5.1. Model estimation and convergence were as described in section 5.2.5. The three-level models allow for the cross-network transfers described in chapter 2 as the

network effects are specified in the μ_j and weighted accordingly (see equation 4.6)

5.4.5 Presenting results

In addition to the results and model fit statistics presented in 5.2.6, I investigated patterns by transfer rate, considering proportion transferred, transfers in and out, and transfers within 24 hours. The rationale for this is that any sensitivity to different weights is most likely to be seen in NNUs with the highest transfer activity, and analysis in chapter 2 showed differences between early and late transfers, with early transfers generally being for a higher level of care.

5.5 Summary

In this chapter I described the statistical methods for the main analyses of the thesis.

First I set out a series of models to investigate some of the statistical issues raised by the research question before considering transfers. In particular I sought to compare differences between modelling clustering by NNU only, by NNU and neonatal network, and ways of modelling differences across NNU levels. I also aimed to assess sensitivity to choice of prior distributions. Then I described a simulation study to assess how sensitive parameters of interest are under a variety of data scenarios and analysis models. I compared the usual weight applied in MM models (proportional with time spent in each cluster i.e. length of stay) with weighting more heavily on the earlier part of stay in line with the findings of chapter 2, and with allocation to first provider to reflect standard practice in the profiling literature. Finally I described the application of MM models with a series of different weights to the neonatal data, building on the models developed in the non-transferred infants. In addition to the weights applied in the simulation study, I weighted the length of stay at each NNU according to the category of care provided, and a simple measure assigning infants to each NNU equally. The results of applying the methods of this chapter are presented for the whole population in chapters 6 and 7. A case study for a single NNU is presented in chapter 8 and any methods pertaining solely to that chapter are described there.

Chapter 6