States have two strategies to increase Medicaid home care policy generosity: expand access to home care programs (i.e. extensive margin) and/or increase the amount of services provided to users (i.e. intensive margin). We refer to these strategies as increasing generosity in the Participation and/or Intensity dimensions of state Medicaid home care policy. This study proposes a new way to measure these two dimensions as latent variables.
Using these measures, we describe trends and compare states in their generosity in both dimensions in 1999 and 2012. Overall in the US, the tendency has been towards an increase in generosity in both dimensions. This may be explained by several factors.
Rebalancing of the LTC market towards home care may translate into higher home care Participation as well as Intensity, as more users with high levels of care needs may move to or stay in the community. Globally, the increase in the number of Waiver programs over the sample period translates into more generous eligibility criteria and, consequently, higher Participation. This is the woodwork effect, whereby persons who did not receive home care before are drawn into Medicaid home care programs because of looser eligibility criteria. Besides, more states started to provide personal care under Medicaid Personal Care programs. This may partly explain the increase in home care Intensity, as personal care services are usually provided over a longer period of time than home health services
—i.e. personal care is more intensive. States’ generosity may continue to increase as a result of the ACA provisions. Continued efforts to rebalance LTC towards HCBS are likely to result in further increases in Medicaid home care Participation, through expanded eligibility for Medicaid home care. Moreover, if future expansions target specifically individuals with high LTC needs, Medicaid home care Intensity may continue to increase.
In 2012, sixteen states still have relatively low Participation and Intensity (i.e. negative values of the latent dimensions; highlighted in gray in Table 2.5.6). Of those sixteen states, Arizona, New Mexico, Utah, and Wyoming don’t participate in any of the main HCBS programs introduced or expanded in the ACA (HCBS Plan Option, Community First Choice, Balancing Incentive Program, and Money Follows the Person). Yet, they all provide HCBS via non-Medicaid funds, not considered here (Mollica et al., 2009).
MSIS data provide useful indicators for the analysis conducted here. However, they may be prone to some errors, e.g. in the process of aggregation to the state level (KFF, 2004). Two actions reduced the influence of erroneous data. First, negative expenditure amounts, which may occur due to adjustments to prior year expenditures, were replaced by missing. Second, when decompositions of users or expenditures by age groups had proportions of ‘unknown age group’ larger than 5%, values were replaced by missing too.7 Latent variables allow us to measure state Medicaid home care policy dimensions comprehensively. They are especially useful to analyze trends in states’ generosity because they work as summary indices. They can also be useful to assess the effects of Medicaid home care policy generosity on various outcomes, as a way to deal with measurement error from two sources: arbitrarily choosing an observed indicator that provides a partial view of a dimension, and measurement error in the indicator itself. However, to use latent variables requires a rich dataset with many indicators. Besides, estimated factor scores
—i.e. values of a latent dimension— must be used with caution in regression. Using them as explanatory variables generally gives biased regression coefficients (Chapter 3).
Lastly, results based on latent variables may be difficult to communicate, because latent variables have no scale and the same change in home care policy may be achieved through different combinations of changes in the indicators. When using observed indicators is preferred, factor analysis can help motivate their selection, by seeing which indicators have the largest loadings. Latent variables can be used to measure home care policy in other countries where it is also decentralized, such as Canada or Switzerland, or other policies —e.g. nursing home policy.
Contrasting the Participation and Intensity dimensions of Medicaid home care policy may deserve more attention in the future. Increasing the extensive and/or intensive margins of home care use is an important choice made by states that, conditional on budget constraints, may entail tradeoffs. As the ACA progresses, it will be important to monitor Medicaid home care Participation and Intensity across the US states, using a latent variable approach, and analyze their impacts on diverse outcomes.
7Due to these two issues, depending on the indicator, there are between 0% and 33% missing values.
As we use the full information maximum likelihood estimator, no observations are dropped.
Table 2.5.6. States’ generosity in Medicaid home care policy Participation and Intensity in 2012 and participation in main ACA HCBS-promoting programs
States Estimated values in 2012a ACA HCBS programs
Participation Intensity HCBS Plan
Alabama 0.39 -0.84 No No No Yes
Alaska 1.87 1.82 No No No No
Arizona -0.91 -0.72 No No No No
Arkansas 0.45 -0.18 No Yes Yes Yes
California 2.59 0.18 Yes Yes No Yes
Colorado -0.48 0.92 Yes No No Yes
Connecticut 1.20 0.32 Yes No Yes Yes
Delaware -0.81 -0.35 No No No Yes
District of Columbia 3.34 -0.03 Yes No No Yes
Florida -0.85 0.34 Yes No No No
Georgia -0.88 -0.80 No No Yes Yes
Hawaii -0.27 2.72 No No No Yes
Idaho -0.16 0.25 Yes No No Yes
Illinois -0.51 0.21 No No Yes Yes
Indiana -0.70 1.66 Yes No Yes Yes
Iowa -0.02 -0.23 Yes No Yes Yes
Kansas -0.78 -0.52 No No No Yes
Kentucky -0.59 -0.62 No No Yes Yes
Louisiana 0.10 0.34 Yes No Yes Yes
Maine -0.65 -0.44 No No Yes Yes
Maryland 0.00 2.25 Yes Yes Yes Yes
Massachusetts 0.32 1.15 No No Yes Yes
Michigan 1.44 -0.23 Yes No No Yes
Minnesota 1.02 -0.04 No Yes No Yes
Mississippi -0.58 -0.68 Yes No Yes Yes
Missouri 0.97 0.00 No No Yes Yes
Montana -0.14 0.44 Yes Yes No Yes
Nebraska -0.48 -0.14 No No Yes Yes
Nevada 0.04 0.43 Yes No Yes Yes
New Hampshire -0.69 -0.30 No No Yes Yes
New Jersey 0.41 0.17 No No Yes Yes
New Mexico -0.46 -0.81 No No No No
New York 1.79 1.06 No Yes Yes Yes
Continued on next page...
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... table 2.5.6 continued
North Carolina 1.28 -0.16 No No No Yes
North Dakota -0.63 0.47 No No No Yes
Ohio -0.52 0.08 No No Yes Yes
Oklahoma 0.71 -0.35 No No No Yes
Oregon -0.60 0.04 Yes Yes No Yes
Pennsylvania -0.79 0.31 No No Yes Yes
Rhode Island 0.24 0.36 No No No Yes
South Carolina 0.26 0.23 No No No Yes
South Dakota -0.11 -0.19 No No No Yes
Tennessee -0.45 -0.95 No No No Yes
Texas 0.38 -0.57 No Yes Yes Yes
Utah -0.69 -0.15 No No No No
Vermont 0.09 -0.43 No No No Yes
Virginia -0.19 0.70 No No No Yes
Washington 0.16 0.49 No Yes No Yes
West Virginia -0.21 0.04 No No No Yes
Wisconsin -0.96 -0.32 Yes No No Yes
Wyoming -0.82 -0.15 No No No No
Gray highlights states where both Participation and Intensity are below the 1999-2012 sample average. aFrom Table 2.4.3. 2012 or last year available (2010 for two states and 2011 for twenty-five states). bParticipation status as of March 2015. cParticipation status as of December 2014. dParticipation status as of November 2014. Source: The Henry J. Kaiser Family Foundation State Health Facts (KFF, 2015)
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2.A Model of canton home care Participation and Intensity in Switzerland
The analysis of state Medicaid home care Participation and Intensity is replicated for canton home care Participation and Intensity in Switzerland. Here, we present the main results of that analysis.
We use data from the Swiss Home Care Survey, which covers the twenty-six cantons between 1997 and 2012 (416 observations). The indicators considered are identical to those used to measure state Medicaid home care Participation and Intensity in the US, except that hours are used instead of expenditures. This is because the decomposition of expenditures by service type is not available in the Swiss dataset. Users of both home health and personal care services in a given year appear twice in the total home care users count, as happened with MSIS data for the US. The methods employed here are the same as the ones used to measure state Medicaid home care policy. Thus, below we show only the results of the final CFA model that is applied to all cantons.
Table 2.A.1 displays the summary statistics of the six indicators of Participation and Intensity in the final model. All indicators vary considerably across cantons and over time.
For example, the proportion of 65+ persons that use home care varies between about 2%
in Ticino in 1997 and 34% in Geneva in 1998.8 Appenzell Innerrhoden provided less than 11 hours of home care per user in 1999, whereas home care users in Ticino in 1997 received almost 100 hours of care, on average.
The final model of canton home care Participation and Intensity is presented in Figure 2.A.1. It has excellent fit —e.g. RMSEA=0.02, TLI=0.99. The correlation between the two dimensions is set to zero, as in the US model —i.e. they are independent.
In the Swiss model, home health users per capita replaces personal care users per capita (US model) as a measure of the Participation dimension. This dimension accounts for 71% of the variation in home health users per capita (0.842), 96% of the variation in home care users per 65+ persons (0.982), and 61% of the variation in home care users per LTC
8This number is likely to be inflated by double counting in total home care users.
Table 2.A.1. Summary statistics of indicators of canton home care Participation and Intensity
Mean Std. Dev. Min. Max.
Participation indicators
HH users per capita (%) 1.94 0.57 0.13 3.62
(TI, 1997) (GL, 2010)
HC users per 65+ persons (%) 16.12 4.85 1.82 33.54
(TI, 1997) (GE, 1998)
HC users per LTC user (%) 73.40 7.19 22.85 87.28
(TI, 1998) (JU, 2002) Intensity indicators
HH hours per user 44.48 12.74 10.18 88.08
(AI, 1997) (JU, 2004)
HC hours per user 43.60 10.53 10.82 99.65
(AI, 1999) (TI, 1997)
HC hours per 65+ user 46.81 11.25 10.04 107.69
(AI, 1999) (TI, 1997)
Canton and year in parentheses. HH = home health care; HC = total home care; HC = HH + personal care; LTC = HC + nursing home care. Canton abbreviations: AI = Appenzell Innerrhoden; GE = Geneva; GL = Glarus; JU = Jura; TI = Ticino.
user (0.782). Home care users per 65+ persons has the largest loading and home care users per LTC user has the smallest loading in the Participation dimension, as happens in the US model. For the Intensity dimension, there are also similarities between the two models: home health hours per user, in the Swiss model, and home health expenditures per user, in the US model, have the smallest loadings among the three Intensity indicators.
Here, the Intensity dimension accounts for 100% of the variation in home care hours per 65+ user, because the loading of that indicator is set to one, to avoid a Heywood case
—i.e. negative estimated residual variance (not statistically significant). Overall, despite the differences between the two countries, the similarities between the two sets of results add external validity to our model of home care policy generosity. Cronbach’s alpha takes values 0.90 for the Participation dimension, and 0.92 for the Intensity dimension; i.e. the Swiss model has excellent internal consistency.
The values of canton home care Participation and Intensity are presented in Table 2.A.2, for the years 1997 and 2012. Those values are estimated based on the model in Figure 2.A.1. As a reminder, the values are standardized over all 416 canton-year observations. Thus, negative values indicate a level of generosity in the Participation or Intensity dimension below the sample average; the contrary for positive values. Home care Participation increased in twenty of the twenty-six cantons between 1997 and 2012, while home care Intensity increased in eight cantons (2012 values highlighted in gray). In
Figure 2.A.1. Model of canton home care Participation and Intensity: confirmatory factor analysis
HH = home health care; HC = total home care. aCorrelation between Participation and Intensity set to zero. bLoading set to one. Standardized loadings. All loadings statistically significant (p<0.01). Sample includes all cantons. Model fit:
χ2=19.24 (p=0.26), RMSEA=0.02, CFI=0.99, TLI=0.99, and SRMR=0.11.
2012, the levels of generosity in the Participation and Intensity dimensions are below the sample averages in nineteen cantons and ten cantons, respectively (i.e. negative values).
Overall between 1997 and 2012, the prevalent trend was an increase in Participation and a decrease in Intensity; this took place in sixteen cantons. The opposite occurred in four cantons where Participation decreased and Intensity increased. Only four cantons increased generosity in both home care policy dimensions. Z¨urich and Geneva were less generous in both dimensions in 2012 than in 1997.
The tendencies identified in Table 2.A.2 are easily seen in Figures 2.A.2 and 2.A.3.
These figures map the different degrees of generosity in canton home care Participation and Intensity in 1997 and 2012. Darker shades of gray represent higher quartiles of the 1997-2012 distribution of generosity. For the Participation dimension, the 2012 map is much darker than the 1997 one —i.e. many cantons with a darker shade. The largest increase in generosity in this dimension happened in Zug, which moves from the bottom to the top quartile. Four other cantons move from the first to the third quartile or from
Table 2.A.2. Estimated values of home care Participation and Intensity
Cantons Participation Intensity
1997 2012 1997 2012
Z¨urich -0.07 -0.30 0.15 -0.44
Bern 0.09 0.71 0.53 0.55
Appenzell Ausserrhoden -0.37 -0.02 -1.18 -0.33
Appenzell Innerrhoden 0.37 -0.00 -2.98 -1.03
St. Gallen -0.22 -0.01 1.08 -0.21
Graub¨unden -0.32 0.05 2.30 0.48
Aargau -0.90 0.07 0.24 -0.48
Thurgau -0.25 0.23 -0.44 -0.28
Ticino -2.53 0.24 5.20 -0.07
Vaud 0.45 1.25 0.30 -0.58
Valais 0.16 0.65 -0.30 -0.75
Neuchˆatel 0.80 0.71 -1.59 -0.20
Geneva 2.55 0.99 0.72 0.27
Jura 1.65 1.75 1.80 1.03
Cantons in federal order. Gray cells highlight values that are higher in 2012 than in 1997.
the second to the fourth —i.e. they are at least two shades darker in 2012 than in 1997.
Obwalden is the only canton with a marked decrease in generosity, moving from the top to the bottom quartile. In 2012, generosity in the Participation dimension was specially high in the Lake Geneva and Mittelland regions —i.e. Western half of Switzerland, including all French-speaking cantons. The Central region was the least generous in this dimension (e.g. Obwalden, Nidwalden).
For the Intensity dimension, the 2012 map is lighter than the 1997 one —i.e. many cantons with a lighter shade. Five cantons move from the third to the first quartile or from the fourth to the second between 1997 and 2012 —i.e. they are two shades lighter in 2012 than in 1997. Uri, Schwytz, and Zug move from the top to the bottom quartile.
There are no marked increases in generosity in this dimension; no canton moves up by more than one quartile. Globally in 2012, the Mittelland region was the most generous in the Intensity dimension and the Central region was again the least generous.
Figure 2.A.2. Maps of generosity in canton home care Participation in 1997 and 2012
(a) 1997
(b) 2012
Darker shades represent higher Participation; the 1997-2012 distributions are cut into quartiles.
In sum, in most cantons between 1997 and 2012, generosity in the Participation dimension of home care policy increased, while generosity in the Intensity dimension decreased. This indicates that more people got access to home care services, while users received less care, on average. In 2012, the Mittelland region is the most generous on both dimensions, and the Central region is the least generous.
Finally, Tables 2.A.3 and 2.A.4 show the cantons ordered by decreasing generosity in the latent dimensions and the observed indicators. We report 2010 values because this is the last year in which all indicators are available. Two facts are worth noticing. First,
Figure 2.A.3. Maps of generosity in canton home care Intensity in 1997 and 2012
(a) 1997
(b) 2012
Darker shades represent higher Intensity; the 1997-2012 distributions are cut into quartiles.
as happens in the US case, some observed indicators may give a different picture from the one we get when looking at the latent dimensions. Second, the canton rankings by each dimension are the most correlated with the rankings by the two indicators with the largest loadings: home care users per 65+ persons, for Participation, and home care hours per 65+ user, for Intensity. This is also the case in the US model. Though choosing a single indicator gives an imperfect view of a dimension, when such strategy is preferred, choosing the best indicator available is very important.
Overall in Switzerland, between 1997 and 2012, the tendency has been an increase in
Table 2.A.3. Comparison of canton rankings based on the Participation latent dimension and observed
Appenzell Innerrhoden 10 11 11 8
Appenzell Ausserrhoden 11 9 10 25
Graub¨unden 12 14 13 11
Z¨urich 13 18 12 17
Correlation with Participation 0.88 1.00 0.55
a2010 because latest year HC users per 65+ persons is available. Cantons ranked by decreasing generosity. HH = home health; HC = total home care.
generosity in the Participation dimension of home care policy and a decrease in Intensity.
The expansion in the number of people with LTC needs, in part due to population aging, party explains the increase in Participation. Another contributing factor may be more widespread information about the existence of home care services. On the supply side, efforts to rebalance LTC towards home care and away from nursing homes may also explain higher Participation levels. Two factors may help explain the decrease in Intensity.
First, over time there has been an effort to better allocate patients to the appropriate LTC setting —i.e. persons with more intensive needs go to nursing homes. Second, the development of post-acute home care means that a larger proportion of home care users are short-term patients —i.e. low-intensity users.
To conclude, we have similar models of the Participation and Intensity dimensions
Table 2.A.4. Comparison of canton rankings based on the Intensity latent dimension and observed
Appenzell Ausserrhoden 21 19 20 21
Geneva 22 12 25 22
St. Gallen 23 24 17 23
Neuchˆatel 24 20 23 24
Appenzell Innerrhoden 25 23 26 25
Zug 26 26 24 26
Correlation with Intensity 0.86 0.83 1.00
a2010 because latest year HC hours per 65+ user is available. Cantons ranked by decreasing generosity. HH = home health;
HC = total home care.
of home care policy for the US states and the Swiss cantons. This further validates our measurement strategy. We find different trends in generosity in the two countries.
In the US since the late nineties, generosity in both the Participation and Intensity dimensions has increased in most states. In Switzerland, we observe an increase in home care Participation and a decrease in home care Intensity in the majority of cantons.
Chapter 3
Dealing with bias in linear regression with factor scores: OLS-corrected
and 2SLS estimators
Abstract
Latent variables are sometimes included as explanatory variables in regressions. Researchers typically use factor analysis to model and attribute values to those variables —i.e. factor scores.
The use of factor scores, as if they were observed variables, causes all regression coefficients to be biased and inconsistent —an issue that has been neglected so far. Our work has three theoretical contributions. First, we show that OLS estimation of linear regression with factor scores gives biased and inconsistent estimates due to measurement error. Second, we derive a consistent OLS-corrected estimator. Third, we formulate the 2SLS estimator, which deals with endogeneity from additional sources besides measurement error —i.e. simultaneity, unobserved heterogeneity. We compare the OLS, OLS-corrected, and 2SLS estimators using simulated data and show the impact of using invalid or weak instruments on consistency of 2SLS. Finally, as an illustration with real-life data, we conduct an empirical study of the effect of home care policy, measured as a latent variable, on observed nursing home use rates. We conclude with a guide on how to proceed to estimate linear regression with latent explanatory variables.
Keywords: Factor analysis; Regression with factor scores; Measurement error bias;
Endogeneity; Simulation; Instrumental variables; Home care policy; Nursing home use JEL classification: C15; C26; C38; H75; I18
This paper is co-authored with Jaya Krishnakumar and France Weaver. We appreciate the comments of Eva Cantoni, Tamara Konetzka, and the participants at the Economics and Finance Seminar at the University of Neuchˆatel.
3.1 Introduction
Latent variables are used to measure concepts that are not directly observable. One way to do that is by modeling a number of observed correlated indicators as functions of a
Latent variables are used to measure concepts that are not directly observable. One way to do that is by modeling a number of observed correlated indicators as functions of a