ONLINE TECHNICAL APPENDIX



APPENDIX/SUPPLEMENTAL DIGITAL CONTENT

Nurse Staffing and Quality of Care With Direct Measurement of Inpatient Staffing

This appendix provides some additional information on the sample, model, calculation of marginal effects, and conversion of nurse staffing measures that could not be in included in the manuscript due to the word limits for the Brief Report article type.

Additional notes on the sample and the model.

We focused on staffing at the inpatient level, hence we only included staffing at inpatient acute cost centers; these "Daily Hospital Services" acute care cost centers included medical/surgical intensive care, coronary care, pediatric intensive care, neonatal intensive care, psychiatric intensive (isolation) care, burn care, other intensive care, definitive observation, medical/surgical acute, pediatric acute, psychiatric acute for both adult and adolescent and child, obstetrics acute, other acute care, nursery acute, and other daily hospital services.

As indicated in the paper, we excluded observations when adjusted daily census was less than 20 and when the expected number of adverse outcomes less than 15. Since the model included a lagged dependent variable as a regressor and the model was estimated after applying the first-difference transformation, however, we judged it appropriate to apply these exclusion to both the current year's observations and previous observation. For example, an observation would be excluded if the current year's or the previous year's expected number of adverse outcomes was less than 15.

The dependent variable was a standardized ratio, implying that the error variance depended on the number of expected and observed adverse events (mortalities or FTRs). The standard error for the standardized ratio equals [pic]. To normalize the error variance, we weighted the data by the mean of the inverse of the standard error for this ratio. By applying the same weight across years for a given hospital, we assure that the variation for a given hospital across years was due to changes in the variables, not the weights.

The analytical approach followed that in a previous study by Mark et al. (2004). There was one difference in the application of the Arellano and Bond (1991) estimator relative to the previous study. If the error term is autocorrelated, then lagged values of the dependent variable are not valid instruments for the lagged first-difference of the dependent variable. Hence, Arellano and Bond propose a test for second order serial correlation in the residuals; if the error term in levels is not serially correlated then residuals in the first differenced model should not exhibit second order serial correlation. Mark et al. (2004) applied this specification test and did not observe second order serial correlation. But we did observe statistically significant second order serial correlation for the mortality ratio. To address this problem we made weaker assumptions about which instruments are valid. For example, when no serial correlation is present, the twice lagged value of the dependent variable, (yi,t-2), is a valid instrument for the first difference of the lagged dependent variable, (yi,t-1 – yi,t-2). If there is serial correlation resulting from a moving-average process of order one, however, yi,t-2 is no longer a valid instrument but yi,t-3 is valid. We note that an alternate method to address the problem of serial correlation is to add another lag of the dependent variable as a regressor, that is, specify that quality of care in time t is a function of quality of care in time t-1 and quality of care in time t-2. Although these two ways of addressing serial correlation are quite different, we note that they had little consequence for the estimates of greatest interest in our study: the estimates of marginal effects in Table 3 of the paper are little changed when we add the twice lagged value of the dependent variable as a regressor instead of making weaker assumptions about which instruments are valid.

Regression coefficient estimates are reported in Table A1 of this appendix.

Notes on the calculation of marginal effects in Table 3 of the manuscript.

Table 3 illustrates the marginal effects of RN staffing implied by our estimates. As we indicate in the paper, since the regression specification included RN staffing level, the square of RN staffing level, and the interaction of RN staffing level with LVN staffing level and with Aide staffing level; the marginal effect of a one FTE per 1,000 inpatient days increase in RN staffing depends on the level of RN staffing as well as on the levels of LVN and Aide staffing. To explain further, consider equation (1) which gives the specification for the mortality ratio (except that, for simplicity, the variables for hospital and market characteristics are suppressed).

(1) Mortality Ratioit = αi + β1Mortality Ratioit-1 +

β2 RN FTEs per 1,000 Inpatient Daysit +β3(RN FTEs per 1,000 Inpatient Daysit) 2 +

β4LVN FTEs per 1,000 Inpatient Daysit + β5 (LVN FTEs per 1,000 Inpatient Days it) 2 +

β6 Aide FTEs per 1,000 Inpatient Daysit + β7 (Aide FTEs per 1,000 Inpatient Days it) 2 +

β8(RN FTEs per 1,000 Inpatient Daysit × LVN FTEs per 1,000 Inpatient Daysit) +

β9(RN FTEs per 1,000 Inpatient Daysit × Aide FTEs per 1,000 Inpatient Daysit) +

β10(LVN FTEs per 1,000 Inpatient Daysit × Aide FTEs per 1,000 Inpatient Daysit) +

… + u it

In (1), subscript i indicates the hospital (so, for example, αi is the fixed effect for hospital i) and subscript t indicates the year. Given (1), the marginal effect of a one RN FTE per 1,000 inpatient days increase in staffing on the mortality ratio equals

(2) β2 + β3{(RN FTEs per 1,000 Inpatient Days +1) 2 - (RN FTEs per 1,000 Inpatient Days ) 2} + β8 LVN FTEs per 1,000 Inpatient Days + β9 Aide FTEs per 1,000 Inpatient Days)

It is clear from Equation (2) that calculation of a marginal effect requires specifying a the level of RN staffing, LVN staffing, and aid staffing. The first entry of Table 3 in the manuscript provides the marginal effect estimate when RN FTEs per 1,000 inpatient days increase from 2.97 to 3.97 and LVN FTEs per 1,000 inpatient days are 0.32 and aide FTEs per 1,000 inpatient days are 0.91. In this circumstance, equation (2) simplifies to β2 + β3{3.97 2 – 2.972} + β8 0.32 + β9 0.91 = β2 + β3 6.94 + β8 0.32 + β9 0.91. That is, the marginal effect is a single, linear combination of the parameters. We used the Stata (Version 10.1) command –lincom- to calculate the standard errors for the linear combination of parameters making up the marginal effects reported in Table 3.

Notes on calculation of marginal effects reported in Table 4 of the manuscript.

As indicated in the paper, Table 4 compares results from the study reported in this manuscript to four, large multi-hospital studies that measure patient outcomes using mortality or FTR. These studies variously measured staffing as a patient to RN ratio, number of direct care hours, or FTEs per 1,000 inpatient days. We assumed that a patient to nurse ratio of 6 (over a 24 hour day) implied 4 (= 24/6) RN direct care hours per patient day, a patient to nurse ratio of 5 implied 4.8 (=24/4) RN direct care hours per patient day, and so on. To translate between RN FTEs per 1,000 inpatient days we assumed that 87.5% of total paid hours were productive (direct care) hours. Assuming 2,080 paid hours in a year for an FTE, implies .875 ( 2,080 = 1,820 productive hours per FTE and 1,820/1,000 = 1.82 productive hours per thousand inpatient days. Hence, RN hours per patient day equal 1.82 ( RN FTEs per 1,000 inpatient days, or, equivalently, RN FTEs per 1,000 inpatient days = (1,000/1,820) ( RN hours per patient day. For example, a nurse to patient ratio of 6 corresponds to 4 RN hours per patient day and 2.20 RN FTEs per 1,000 inpatient days.

Table A1. Complete Set of Regression Estimation Results for Quality of Care Measures.

(Marginal Effect of a One Hour per Patient Increase in RN Staffing Reported in Table 3.)

| | |Failure-to-Rescue |

| |Mortality |Ratio |

|Variable |Ratio | |

| | | |

|Dependent Variablet-1 |0.272*** |-0.115 |

| |(0.081) |(0.077) |

|RN FTEs per 1,000 Inpatient Days |-0.131 |0.020 |

| |(0.070) |(0.144) |

|RN FTEs per 1,000 Inpatient Days 2 |0.005 |-0.018 |

| |(0.009) |(0.018) |

|LVN FTEs per 1,000 Inpatient Days |-0.472 |-0.177 |

| |(0.245) |(0.557) |

|LVN FTEs per 1,000 Inpatient Days 2 |0.001 |-0.114 |

| |(0.063) |(0.146) |

|Aide FTEs per 1,000 Inpatient Days |-0.036 |0.044 |

| |(0.108) |(0.193) |

|Aide FTEs per 1,000 Inpatient Days 2 |-0.001 |-0.017 |

| |(0.026) |(0.058) |

|RN Staffing × LPN Staffing |0.125* |0.126 |

| |(0.052) |(0.110) |

|RN Staffing × Aide Staffing |0.018 |0.012 |

| |(0.027) |(0.053) |

|LPN Staffing × Aide Staffing |0.042 |-0.088 |

| |(0.077) |(0.196) |

|Operating Margin |-0.001 |-2.2e-04 |

| |(0.001) |(0.002) |

|Case-mix Index |-0.073 |0.089 |

| |(0.113) |(0.181) |

|Saidin Index |0.017 |0.007 |

| |(0.011) |(0.017) |

|System |0.097 |0.054 |

| |(0.053) |(0.093) |

|Public |-0.293 |-2.389 |

| |(0.547) |(3.357) |

|For-profit |-0.163 |-0.157 |

| |(0.148) |(0.274) |

|Payer Mix |0.128 |0.241 |

| |(0.078) |(0.166) |

|Number of Beds |-2.7e-04 |-0.001 |

| |(2.5e-04) |(3.8e-04) |

|Hospital Use |2.1e-04 |4.2e-04 |

| |(1.3e-04) |(3.6e-04) |

|Herfindahl Index |0.377 |0.820 |

| |(0.421) |(0.937) |

|Number of HMOs |0.003 |0.038* |

| |(0.008) |(0.017) |

|HMO Penetration |-0.006 |0.422 |

| |(0.238) |(0.622) |

|HMOs × HMO Penetration |-0.009 |-0.068* |

| |(0.014) |(0.032) |

|Dummy Variables for Years |( |( |

| | | |

|Number of Observations |1,073 |670 |

a Standard errors (in parentheses) beneath the estimates of the marginal effects.

* Significant at the 0.05 level;

** significant at the 0.01 level;

*** significant at the 0.001 level.

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