The impact of long‐term care on quality of life

Abstract Long‐term care services are provided to help people manage the consequences of impairment, but their impact goes beyond the meeting of basic needs. Accordingly, the main aim was to explore the marginal effectiveness of care when measured in terms of people's overall care‐related quality of life (CRQoL) and assess changes in marginal effect for increasing intensity. The associated aim was to refine and apply an observational method to estimate marginal effectiveness. A “production function” approach was used with survey data, including Adult Social Care Outcomes Toolkit‐measured CRQoL, whereby we statistically modelled the expected relationship between service utilisation rates and CRQoL. This method seeks to limit endogeneity issues by controlling on observables and using instrumental variable. Using a survey of publicly funded long‐term care service users in England, we found that community‐based long‐term care significantly improved people's CRQoL but with diminishing marginal effects and effects differentiated by baseline impairment levels. There are implications for how the care system should respond to changes in global public budgets. For example, where there is unmet need, a system aimed to maximise (unadjusted) CRQoL would put more emphasis on access (more recipients) than intensity of support compared to a system operating on a needs basis.


| Service user utility
The utility of a person (i) with care needs-denoted by U i -is likely to be affected by the severity of their condition and other needs-related circumstances (u n i ), the use of informal care (Y i ), and the amount of care services they utilise. These services can be publicly funded (X L i ) or privately paid (X P i ). Following Becker's model of altruism in the family (Becker, 1981), the service user's utility will also be affected by the utility of informal carers (V i ) and vice versa. Finally, the person will receive utility from non-care aspects of their lives (u m i ). Taking these elements together gives Similarly, the utility of any carer of person i would be The amount of formal services the person uses is determined by a "needs" assessment, which in England is generally undertaken by a local authority care manager who works with the service user and their family to develop a care plan. The level of publicly funded support available is affected by the eligibility policy of the assessing authority. We denote L i as the policy of the local authority responsible for person i, where the assessment results will depend on needs, and the family's resources, denoted by G i . Thus, the amount of service use is given by the vector function: X L i L i ; u n i ; G i À Á , which can include a number of different service components (e.g., some home care and some day care). This variable can be regarded as being exogenous.
The amount of both informal carer support and privately paid care will be determined by a "family assessment" that will account for the outcome of the public care services assessment, X L i . In particular, we can define The function Y Ã i can be used in 1, after solving for V i , to give a partial reduced-form function:

| Empirical analysis
This utility function 3 is the basis for an empirical model of person i's current CRQoL, which we denote by u c i . Because we focus on CRQoL, u m i and v m i are constants that we can set to zero. Taking a linear approximation of 3, we have where X i ¼ X L i þ X P Ã i , that is, the total amount of formal services being used. Also, u e i ¼ θ 0 þ θ n u n i þ θ v v n i þ θ G G i is the person's CRQoL in the absence of services (and where informal care support is at its level when no formal services are used, because the function θ implicitly embodies changes in informal care for different (exogenous) provision of formal services X i ). Finally, ϵ uc i is the error term. In practice, people with care needs will be using services, and therefore, u e i is a hypothetical level of CRQoL that could be expected if formal services were not used. For shorthand, we refer to u e i as the "expected CRQoL." As a simplifying assumption, we can define a function so that the effect of service utilisation X i is linear in a coefficient β u , that is, Using an explicit functional form requires us to make an assumption about the shape of the x i function. Substituting into 4 gives, One approach would be to estimate 5 using ordinary least squares (OLS) with a range of observable needs-related factors as proxies for u n i and v n i . However, in practice, we only have an imperfect observation of all the needs-related factors in u n i and v n i , and so in u e i , leaving unobservable needs factors in the error term that are also correlated with service levels x i u e i À Á . This omitted variables endogeneity bias arises because the relevant unobserved characteristics of people providing the counterfactual experience (of a different level of utilisation of services) might be different to those of people using services of amount x i (X i ). Another approach to estimate effect sizes is using self-estimation of the counterfactual (Mueller, Gaus, & Rech, 2014). Service users are asked to estimate their own counterfactual experience of a different level of service use. Practical applications of this approach have involved asking people to rate their CRQoL in the absence of services (i.e., where X i = 0), which produces an estimate of u e i . Mueller et al. (2014) recognise that because individuals have to hypothesise about the counterfactual state, some "self-estimation bias" is possible.
This approach can be combined with (OLS) regression on observables whereby the self-estimate of u e i is also used as a control factor. To be explicit, suppose the observable needs-related proxies are denoted by the vector n i . We can define e u e i as the self-reported indicator of expected utility: u e i ¼ e u e i þ η e i ¼ e u e i þ β n n i þ η ne i where η e i is the error and η ne i is the error without any observable need effects. Using this in 5 gives In principle, the estimate of u e i embodies some information about non-observable factors, and so this method should help reduce the chance of endogeneity bias, depending on the correlation between self-estimation bias and the level of service use. Unfortunately, the unobserved self-estimation bias η ne i is not likely to be entirely independently distributed with respect to u e i . In particular, the person's self-estimation might be affected by the scale of their prior need and the amount of services they receive. Consequently, the error component η ne i may still be correlated with x i . A further option is to use IV estimation. The challenge is to find good instruments, that is, factors that affect the current CRQoL score (u c i ) but only through service use (x i ), giving an IV specification of x i that is not correlated with u e i .

| Instrumentation
Our theoretical set-up can help guide the choice of instruments. One potential instrument is the local authority's social care policy; the term L i in 5. As a result of policy preferences and exogenous supply factors, we can expect that a person whose care is organised by one local authority (LA) would receive a different level of support compared with a person with the same circumstances whose care is organised by a different LA. Moreover, this difference would not be a function of the individual's characteristics, and so have no bearing on u e i . Rather than use a simple dummy variable for LA in the estimation, we instead created a "spatially lagged" service use variable. This is calculated as the average service utilisation amount of other people (k) in the sample living in the same LA as the person in question: The level of service use by other people in the sample (x j ) should have no direct effect on a given service user's expected CRQoL (u e i ) but is likely to be correlated with their service use (x i ).
We therefore undertook the following IV estimation of 6: where b x i is a function of n i , self-reported indicator of expected utility, e u e i , and the spatial lag S x IV i À Á instrument. The spatial lag instrument seeks to capture differences in policy and supply between care authorities. However, because this variable is partly defined at the LA level, there is a possibility that it might be correlated with factors that vary on a geographical basis, where these factors could have a direct effect on CRQoL. We cannot directly test the validity of the instrument, but following the literature we assess whether the instrument is balanced with respect to the needsrelated characteristics of service users (Rahman, Norton, & Grabowski, 2016). In particular, we split the sample into two groups around the median value of IV and then test whether paired (t test) comparisons of the covariates, n i and e u e i are significantly different between the two groups. Finding no significant difference provides some reassurance as to the validity of the instrument.
In addition, we included area level controls directly in the model: (a) dummy variables for higher level geographies and (b) a spatially lagged specification of the dependent variable. We would expect omitted geographically correlated variables to be (at least partially) captured by these variables rather than left in the error.

| Functional form
The marginal effects of changes in the intensity of provision X i will depend on the shape of the service effect function We experimented with a range of functional forms of 7, including monomial and polynomial functions, the latter including piecewise polynomials in the form of cubic basis splines. The single intensity-variable (monomial) specifications of 7 were estimated using two stage least squares in Stata 14 with a two-step feasible generalised method of moments estimator (to allow for arbitrary heteroscedasticity).
The polynomial specifications were estimated using the predicted value of the (manually instrumented) intensity as the subject variable in the second stage. We estimated two polynomial functions models: a 3-polynomial model with cube-root, linear, and cubed values of service intensity as terms; and a model with (basis) cubic splines of service intensity. In both cases, the predicted value of service intensity was as estimated, by OLS, as follows: The 3-polynomial variant of 7 was then Splines are piecewise polynomial functions that can have a simple form locally but allow global flexibility. We used cubic splines that are piecewise third-order polynomials that pass through a set of control points. In this case, 7 becomes . This function was also estimated using OLS with predicted values of service intensity from a first-stage estimation.

| Hypotheses
Our first hypothesis is supported if we find ∂u c i ∂X i >0 and ∂ 2 u c i ∂ 2 X i <0 for all observed values of X i . Our second hypothesis is supported in general if we find that ∂ 2 u c i ∂X i ∂u e <0 for observed values of X i over the range of values of u e i . However, a more pragmatic test, which we use, is whether for sample relevant average values of expected CRQoL, such that u e1 >u e2 >u e3 (e.g., where these values are means for terciles of the expected utility distribution in the sample). A practical way to estimate the relevant coefficients is to use interaction terms in 7: and compare ∂u c

| DATA
The main data collection in the Identifying the Impact of Adult Social Care study was a survey of community-based social care service users. The sample was drawn either from a non-stratified random sample of the eligible clients in each local authority's social care database or from those respondents from the 2012/2013 Adult Social Care Survey who indicated that they would like to receive information about follow-up research. In total, 14,021 letters were sent out by the councils (13,654) and home care providers (367). One thousand seven hundred thirty return slips were received (12.3% response rate). Of these, 1,441 indicated an interest in participating in the study and met the study inclusion criteria. Nine hundred and ninety valid interviews were in the final dataset: 546 adults with physical disability or sensory impairment, 224 adults with mental health conditions, and 220 adults with learning disability. The majority of people sampled were interviewed face-to-face, whereas about a quarter were interviewed by telephone. Full details of the survey design and characteristics are available (Forder et al., 2015). The information collected included socio-demographic characteristics, service receipt (i.e., publicly funded, selffunded, and informal), well-being, health status, functional ability, control and autonomy, suitability of home and local environment for mobility needs, social contact and support, and participation in social groups (Forder et al., 2015).
This analysis used an (anonymised) sample including those with physical disability or sensory impairment and mental health. We selected people aged 40 or more, giving 729 observations, of which 699 had valid information about CRQoL. There was some missing data on a number of independent variables, with a final sample of 622 cases.
Service use intensity was measured as cost-weighted utilisation of the main community services: home care, day care, meals, social work support, and equipment and adaptations (the total of LA-funded and privately paid care). In this regard, England-mean (gross) unit cost data were available. The cost-weighted utilisation variable had the usual high degree of rightward skew. To this end, we censored all cost-weighted utilisation values of greater than £1,500 per week (more than twice the gross cost of residential care) to be £1,500. Approximately 3% of cases were censored.
The data also included a subset of cases where the actual cost of the LA-supported care packages was available (as supplied by the respective LA). The cost-weighted utilisation variable (for the whole sample) was rescaled by a constant to give it the same mean as the LA package cost variable (plus 10% to allow for individual top-up) for the subset of cases where both (non-zero) figures were available. Individuals are free to purchase additional care privately to top-up their publicly provided package (see Forder & Fernández, 2009, for estimates).
Even with censoring, cost-weighted utilisation was highly skewed. We therefore used transformed values of x i as noted above. Figure 1 shows kernel density plots. A cube root transformation appeared to best approximate the normal distribution.  Figure 2 shows the distribution of the current CRQoL of the sampled service users, measured using ASCOT. The ASCOT indicators are preference-weighted quality of life measures anchored around quality of life states equivalent to being dead (a score of 0) and having a score of 1 if a person chooses the highest levels on each attribute.

| RESULTS
Descriptive statistics of the main variables are given in Table 1. We found the instrument to be well balanced with regard to respondent's characteristics (control variables). Table 2 has the results. Respondent characteristics were virtually identical between the upper and lower value groups of the instrument, allaying concerns that the instrument might be correlated with other, unobserved factors that directly affect CRQoL.

| Monomial function analysis
The main model 7 results are given in Table 3, using the person's current CRQoL score as the dependent variable with cube root-transformed utilisation as the indicator variable (Model 1). We did not use logged dependent variable specifications (including log-log [Cobb-Douglas] forms) in the main model because the CRQoL includes negative values and the logged distribution was further away from the normal distribution compared to the linear version-see Figure 2.
In line with our first hypothesis, we found a significant positive effect of service utilisation on quality of life (CRQoL). The control factors also had the expected signs. The table includes the first-stage estimation results and shows the significance of the instrument and the strong effects of frailty (ADL count and expected CRQoL) and poor health. The instrument did not appear "weak" with an F test exceeding the conventional threshold of a score of 10. We also found strong support for the utilisation variable being endogenous.
Regarding area controls, we found that a number of the dummy variables for higher level geographies were significant in the second stage but not the first. The spatially lagged specification of the dependent variable was not significant at either stage.
Variant specifications were estimated. Table 4 gives results for specifications with different transformations of the utilisation variable, estimated with a generalised method of moments IV model. Although all three specifications were significant, the log form produced the least probability of information loss, measured using both the Akaike information criterion and the Bayesian information criterion. A linear specification was also estimated, but the (censored) high values of utilisation caused problems-see Figure 1. Dropping the outliers produced a better fit, but the Akaike information criterion and Bayesian information criterion were substantially higher than for the non-linear versions. Table 5 gives results with different specifications of the control variables. We estimated models without the area dummies but found little difference. In another specification, we dropped the self-reported health variable for comparison, but this made no qualitative difference to the result.
Finally, we estimated an IV model with absent-service CRQoL as the dependent variable. In theory as outlined above, absent-service CRQoL should not be a causal function of the intensity of service utilisation. Nonetheless, there remains a possibility that people's judgement of absent-service CRQoL is biased by their situation with regard to using services; for example, people with a high intensity of service use might rate their expected CRQoL to be lower (poorer) than people with more modest levels of service use. In this case, however, we found no significant effect.
Notwithstanding the issues with using a log-transformed dependent variables, for comparison a log-log (Cobb-Douglas) version of the main model was estimated. We found a coefficient on (log) intensity of 0.15, and this supports our hypothesis of decreasing returns to scale.

| Polynomial functional forms
The results for the cube powers polynomial and B-splines models are given in Table 6. Regarding the cubic spline model, after some experimentation we estimated a model with 5 knots, set at the minimum and maximum values, and the three tercile points of the predicted value of cube-root service intensity variable after re-transforming it back to the linear.
B-splines were generated using the frencurv command in Stata 14 and included one upper and one lower out-of-range spline. The spline based at a service intensity of zero was omitted as the base category. The specific form of 10 that we estimated was The splines b k b X i were set at £-36, £36, £83, £158, £647, and £1,136 (with the spline at £0 excluded).
The coefficients for the B-splines model can be interpreted as the change (gain) in CRQoL at the listed service intensity as compared to zero utilisation. Note that for these polynomial functions, the standard errors were not corrected for use of (stochastic) predicted values from the first-stage estimation. Figure 3 shows these results graphically. The base cube root specification-as reported in Table 3-is also displayed for comparison. The splines version showed a very similar functional form to the cube powers polynomial, although with smaller effect sizes. All three versions showed a significant diminishing effect in the main, a result which is consistent with our first hypothesis. The more flexible forms suggest a small increasing marginal effect for changes in utilisation intensity above around £300 per week. This intensity is, however, the 90th percentile of the distribution, so the result should be treated with caution.
It is also worth noting that for service intensities above minimal levels, the size of the estimated marginal changes in CRQoL were very similar between model results with alternative functional form assumptions (as in Figure 3). More generally at service intensities of more than around £25 per person per week, marginal effects were very similar in size between models.

| Mediating effects of impairment
As is clear from the first-stage results as reported in Table 3, the intensity of service use is strongly positively correlated with the severity of the person's condition. Regarding our second hypothesis, after some experimentation we used an interaction term variant, 11, of the cube-powers polynomial model, 9, by adding the terms β w d X
The results are summarised in Table 7. An F test rejected the hypothesis that the interaction terms have zero effect (F = 2.77, p = .041). The results are also plotted in Figure 4, with expected CRQoL in the equations set at three values: the mean value for the whole sample (i.e., average need); the mean for people in the lower half of the distribution below the median (i.e., high need); and the mean for people in the upper half of the distribution (i.e., low need). The pattern is consistent with our second hypothesis. We should also note that low-need people use only about half as much service intensity (an average of £90 per week) compared to high-need people (£220 per week). In other words, low-need people are clustered at the left-hand side of the utilisation intensity distribution whereas high-need people are more often further up the distribution.  Note. CRQoL = care-related quality of life; AIC = Akaike information criterion; BIC = Bayesian information criterion.
* 10% significance level, ** 5% significance level, *** 1% significance level.  Regarding our first aim, we found that community-based long-term care in England does significantly improve the carerelated quality of life of service users (CRQoL); marginal effects are generally reduced with additional intensity of support; and people with more severe conditions (higher need) show greater marginal improvements than people with lower need, other things equal.   The second aim was to further explore the feasibility of using a production function approach and, in particular, the viability of using IV estimation to address the endogeneity or selection issues inherent in this method. We found that spatially lagged service intensity appeared to be a good instrument in this case. It was well balanced on observables and was not weak.
There are a number of general limitations of the analysis to discuss. First, although IV is commonly used, it has wellknown sensitivity to specification, particularly of the instruments. Although we have tested different functional forms, a second issue is that this method does rely on assumptions we make in this regard. Choice of functional form will particularly affect extrapolations (of service intensity) in the range with low observation density in the sample. We did not systematically compare functional forms, but the range of forms tested produced similar marginal effect estimates.

| Resource allocation
The results can be used to inform policy about prioritising public funding between different potential care recipients. Much will depend on the exact goals of the care system, but we can consider a number of scenarios where a system with a policy goal of maximising CRQoL would lead to a different allocation of resources compared to a system that allocated on the basis of need. We use the results from Figure 4 above to illustrate a number of example scenarios, with the caveat that we are using point estimates. Specifically, we use those results discounting the increasing effect after the 90th percentile, instead approximating a linear function after the turning point using the improvement value at the 95th percentile as the reference pointsee Figure 5.
To begin with, we can consider a scenario about the allocation of a fixed budget among a given population with care needs divided equally between high-and low-need people equivalent to our sample. Given the same total budget for the sample (£10,026,640 = 310 per week per person × 622 × 52 weeks), the optimal solution (on the basis of our point estimates) gives approximately £190 per week to high-need people and £120 per week to low-need people. Compared to the observed allocation (£220 per week and £90 per week, respectively), this solution does shift more support to the low need group, but the change is relatively small. As such, this result does not provide a strong case for reallocating resources between given high-need and low-need populations.
In the above example, we worked from a given budget and a fixed number of people by need group. Another scenario is where the budget for care services is not fixed but rather set according to the opportunity costs of public expenditure more generally. Specially, suppose there is some externally determined opportunity cost threshold. With a fixed population, we would determine the optimal where the last £1 spent on care gave the same marginal benefit as the alternative; that is, where the latter opportunity benefit term is conventionally expressed as the (inverse) cost of increasing CRQoL by 1 or λ. For example, at a λ threshold of £30,000 per extra CRQoL year gained, the optimal level of funding for the high-need group would be £75 per person per week, and the low-need group would be £50 per person per week (as based on point estimates). Seemingly, this result implies that the budget could be reduced, but it does not account for changes in access and therefore changes in the number of people served overall.