Top PDF Heterogeneity in preferences for primary care consultations: results from a discrete choice experiments

Heterogeneity in preferences for primary care consultations: results from a discrete choice experiments

Heterogeneity in preferences for primary care consultations: results from a discrete choice experiments

Apart from the important impact of some socio- demographic variables and the health status, the characteristics connected to the participants' past experiences seem to have the greatest influence on the involvement level. Thus, our data suggest that preferences for a different involvement level could be relatively controllable by the caregivers, considering that, to a large extent, they seem to depend on the attitude of the GP in the previous visits and to the long- term ‘personal’ relationship between a patient and his/her GP, that is different from the occasional contacts with a specialist. Patients, in Italy, are in fact free to choose their GP at the age of 18 and once they make their final choice they hardly change it [37]. Consequently, this long term relationship, built over the years and most of the times based on reciprocal trust should not require patients’ total involvement during each consultation. The presence of heterogeneity was also confirmed when taking into account information preferences.
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Heterogeneity in individual preferences for HIV testing: A systematic literature review of discrete choice experiments.

Heterogeneity in individual preferences for HIV testing: A systematic literature review of discrete choice experiments.

preferred home testing [22] . Although, home testing is most conve- nient in terms of travel time, con fidentiality may be a concern, partic- ularly among individuals worried about testing HIV-positive. Fear of disclosure to others in the household may be a barrier to home test- ing. For example, female bar workers in Tanzania preferred that their partner does not know about their HIV test [10] . Similarly a DCE in Zimbabwe found that men and AYA preferred individual HIVST distri- bution to batched distribution (kits provided to the whole house- hold). This may indicate concerns about con fidentiality or coerced testing by others in the household. Travel distance to HIV testing was another source of heterogeneity. Overall, participants preferred shorter travel distances to testing services, indicating opportunity costs associated with travel time and transport costs can be a barrier to testing. However, female bar workers and male porters were more willing to travel longer distances for testing, suggesting that the con- venience of closer testing may be outweighed by con fidentiality con- cerns of testing in one ’s own community [10] . Together, these results suggest that while home testing has high acceptability, it is not uni- versally preferred. To reach the first 95 of UNAIDS ambitious targets, policymakers must implement a combination of HIV testing strate- gies that fit the diverse preferences of different sub-populations.
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Valuation of labour market entrance positions among (future) apprentices - Results from two discrete choice experiments

Valuation of labour market entrance positions among (future) apprentices - Results from two discrete choice experiments

In line with the theoretical arguments outlined above, we conclude that the choice of an apprenticeship position and the choice of a first employment after VET are rational decisions influenced by individual demands and resources as well as by institutional constraints, and derive the following hypotheses: Students applying for apprenticeships and apprentices choosing a first job after VET base their decisions on the subjective evaluation of working conditions and wage. We expect beneficial apprenticeship or job characteristics, e.g., higher wage and shorter commuting duration, to increase the SEU of an apprenticeship or employment position, thus raising the probability of a position being chosen (hypothesis 1). In addition, we explore the factors determining the valuation of vocational and apprenticeship attributes. Following the theory of SEU, we investigate the contribution of individual resources and intentions, as well as the labour market context, to the explanation of preference heterogeneity. Following RCT, preferences vary depending on the demands that are posed towards em- ployment. We expect the utility of an employment characteristic to depend on its relevance to the decision makers' situation. Hence applicants intending to pursue further training during their future employment, should take the willingness of a company to finance education into account when choosing between positions, while respondents who do not intend to take up further training after VET should be indifferent to whether or not the employer offers to subsidise it (hypothesis 2). Work expectations regarding income and working conditions are formed through the experience of parents’ occupational situation. We therefore expect the valuation of ap- prenticeship or job attributes, and hence the determinants of choice of an apprenticeship position and employment within a profession, to vary by socio-economic background (hypothesis 3). Following RCT and the current state of research
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Lexicographic Preferences in Discrete Choice Experiments: Consequences on Individual-Specific Willingness to Pay Estimates

Lexicographic Preferences in Discrete Choice Experiments: Consequences on Individual-Specific Willingness to Pay Estimates

Reported in this paper are the results from an empirical study which investigated the implication on WTP of lexicographic decision-making rules. The analysis is conducted on the results from a discrete choice experiment that was conducted in Ireland designed to elicit WTP for a number of landscape attributes. The landscape attributes in question were Wildlife Habitats, Rivers And Lakes, Hedgerows and Pastures. Each of these landscape attributes were depicted with three levels, either A Lot Of Action, Some Action or No Action. Since valuation of landscape improvements can be very subjective, and verbal descriptions can be interpreted differently depending on individual experience, each level of improvement was qualified by means of digitally manipulated images of landscapes. This study also attempted to take stock of the main advances in the areas of multi-attribute stated preference techniques. In particular, a sequential experimental design with an informative Bayesian update, in addressing the heterogeneity of the estimation of the structural parameters of the random utility model the distributions of taste-parameters were bounded to account for the fact that landscape improvement takes the form of an improvement on the status-quo.
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Patients’ preferences for primary health care – a systematic literature review of discrete choice experiments

Patients’ preferences for primary health care – a systematic literature review of discrete choice experiments

Primarily, the chosen attributes and, thus, the pref- erences elicited by the DCE depend on the specific research question. Even if the study objective is the same, the precise issue might differ. Pedersen et al. [29] and Turner et al. [24] for example both aim to assess primary care consultations in general, but while the first assess preferences regarding different organizational characteristics, the latter estimate the relative importance of continuity of care compared to other aspects of primary care. Therefore, unsurpris- ingly, these two studies obtain different results regarding the preferences for primary health care. Pedersen et al. find the attribute “ Waiting time ” as being the most important one and Turner and col- leagues ascertain the process attribute “ Information and explanation ” to be most significant. Their differ- ent research questions may cause a different selection of attributes and consequently different results although the study objectives are the same. In this context, a replication study using the same research question, the same attributes and levels as an existing DCE but comparing different regions and/or populations would be a useful addition to the literature.
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Applying user preferences to optimize the contribution of HIV self testing to reaching the "first 90" target of UNAIDS Fast track strategy: results from discrete choice experiments in Zimbabwe

Applying user preferences to optimize the contribution of HIV self testing to reaching the "first 90" target of UNAIDS Fast track strategy: results from discrete choice experiments in Zimbabwe

The strengths of this study include use of simulations of how LCT could be affected by changes to programme attributes. We also present preferences for the full HIVST cascade. Although DCE preferences are hypothetical, our study was conducted in communities previously exposed to HIVST, so that participant preferences were shaped by their actual experiences. Using the simulation-based RPL to account for unobserved heterogeneity improves the model fit. However, its complex structure is not well-suited for use in simple excel-based decision support sys- tems, where the utilities are manually entered to predict uptake. We rather used the output from the simpler NL model to simulate the impact of variations in LCT services. Table 3 shows that although the RPL has a better statistical fit, the NL is a good approximation. Nevertheless, there are some small Table 5. Change in uptake of simulated linkage programmes compared to base case for the full sample, by sex and HIV testing history (%)
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Discrete Choice Experiments for Health Policy : past, present, and future

Discrete Choice Experiments for Health Policy : past, present, and future

to the government proposal allowing insurers to set a lower level of reimbursement of care by non-contracted providers. At the time of data collection, this proposal was heavily disputed by the medical profession and widely covered by the media. Restricting provider choice was also one of the reasons for the backlash against Managed Care Organizations in the US in the 1990s [53, 54]. A DCE study in the US setting also found that the physician network was the most important attribute for health plan choice, followed by prescription coverage and costs per visit [24]. Earlier Dutch DCEs found a negative WTP of €76 per year for physician choice from a predefined network, and of €137 for choice from a list established by the health insurer, compared to physician choice based on the gatekeeper model (i.e. a model in which general practitioners decide on referral of patients to medical specialists). From these studies it follows that the WTP for free physician choice was positive (€79) [29, 30]. Our results differ from the most recent DCE study that found no effect of the level of provider choice on insurance preferences [33]. Bergrath and colleagues suggested that the insignificance had to do with consumers being not fully aware of all the aspects that play a role in a health care system with managed competition. Our study confirms earlier DCE findings of preference heterogeneity with respect to restrictions on provider choice and premium [29, 30, 33]. Our results indicate that consumers do not value the focus of insurers on quality of care highly. This might indicate that consumers do not expect insurers to play an important role as purchasing agents of high quality care. Instead, consumers seem to rely on their general practitioner to select which are the best care providers to go to [55]. Strengths and limitations
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Who should be prioritized for renal transplantation?: Analysis of key stakeholder preferences using discrete choice experiments

Who should be prioritized for renal transplantation?: Analysis of key stakeholder preferences using discrete choice experiments

Our findings can be considered alongside a number of earlier non-DCE studies. An Australian-based renal study unlike ours adopted a general public perspective [10]. Respondents were found to prioritize long waiters and the young, but had a split verdict over whether to prioritize those with children. Similarly, renal research into African Americans‟ preferences [9] indicates that kidney allocation based upon HLA matching is considered unfair. However, at the same time, African Americans did not want to receive organs with lower survival rates; note since this paper was published (1997) graft survival for poorer matches has improved. More recently, a 2005 Glasgow renal study [11] has used a non-DCE scenario approach to consider allocation of deceased donor kidneys for transplantation. Interestingly, certain findings from this research conflict with our results (i.e. tissue matching was not a major allocation criterion) although, like our findings, the researchers reported that emphasis was placed on prioritizing long-waiters (albeit defined by time on dialysis, not on waiting lists). One DCE study, a 2010 Canadian article on patients with chronic kidney disease [19] has reported that respondents preferred to prioritize kidney transplants on the basis of a „best match‟ rather than „first come, first served.‟ However, in contrast to our study, this particular DCE considered a wide range of attributes relating to CKD in general (including organ procurement and the organization of care) and as such could provide only a very limited indication of preferences for kidney transplant allocation. The DCE included only one attribute relating to kidney transplants (“H ow should deceased donor kidneys for transplantation be allocated for transplantation”) with just two possible levels „best match‟, or „first come, first served.‟ Moreover, unlike our DCE study which furnished respondents with information on the likelihood of kidney transplants being successful for non-favourable matches, it is unclear whether similar information was provided in the Canadian study to ensure fully informed responses.
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Preferences for linkage to HIV care services following a reactive self test: discrete choice experiments in Malawi and Zambia

Preferences for linkage to HIV care services following a reactive self test: discrete choice experiments in Malawi and Zambia

testing (32). Participants preferred to be followed-up after self-testing by the distributor through phone calls, similar to other HIVST studies (33-35). Although, previous literature demonstrated high rates of linkage and retention into care through an active referral, we found that support for linkage was the least important attribute and would have the smallest effect on encouraging self- testers to link (36-38). We found a strong willingness to link into HIV care given the proposed service configurations. A community-based implementation study in Malawi reported that 77% of 16,660 participants shared their self-test results with distributors (6). This underpins the need for HIVST services to provide support for linkage to onward services to facilitate the potentially high demand.
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Differentiated care preferences of stable patients on ART in Zambia: a discrete choice experiment

Differentiated care preferences of stable patients on ART in Zambia: a discrete choice experiment

Background: Although differentiated service delivery (DSD) models for stable patients on antiretroviral therapy (ART) offer a range of health systems innovations, their comparative desirability to patients remains unknown. We conducted a discrete choice experiment to quantify service attributes most desired by patients to inform model prioritization. Methods: Between July and December 2016 a sample of HIV-positive adults on ART at 12 clinics in Zambia were asked to choose between two hypothetical facilities which differed across six DSD attributes. We used mixed logit models to explore preferences, heterogeneity and trade-offs. Results: Of 486 respondents, 59% were female and 85% resided in urban locations. Patients strongly preferred infrequent clinic visits (3 vs. 1-month visits: β (i.e. relative utility) =2.84; p <0.001). Milder preferences were observed for: waiting time for ART pick-up (1 vs. 6 hrs.; β=-0.67; p<0.001) or provider (1 vs. 3 hrs.; β=-0.41; p=0.002); ‘buddy’ ART collection (β=0.84; p <0.001); and ART pick-up location (clinic vs. community: β=0.35; p=0.028). Urban patients demonstrated a preference for collecting ART at a clinic (β=1.32, p<0.001), and although the majority of rural patients preferred community ART pick-up (β=-0.74, p=0.049), 40% of rural patients still preferred facility ART collection. Conclusions: Stable patients on ART primarily want to attend clinic infrequently, supporting a focus in Zambia on optimizing multi-month
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Patients' valuation of the prescribing nurse in primary care: a discrete choice experiment

Patients' valuation of the prescribing nurse in primary care: a discrete choice experiment

8 (details upon request). Prior to full analysis, the estimated models were checked for theoretical validity by considering the sign on the coefficients of the alternative-specific constants and attributes. Table 3 shows the descriptions of the variables included and the a priori hypothesis expected for the sign of each coefficient. The table then describes the arguments in the ‘own doctor’ utility function (Equation 1) and ‘NIP’ utility function (Equation 2), respectively. The labels for ‘own doctor’ ‘prescribing nurse’ alternatives are accounted for as alternative-specific constants (a1 and a2) within each utility function, respectively. We expected respondents to prefer longer consultations and higher levels of quality relating to the patient–professional interaction of their patient experience. In other words positive signs for all attributes were expected. We investigated the impact differences in patient characteristics had choice through a number of hypotheses. For example, we hypothesized that individuals with poorer health may be more likely to choose the ‘own doctor’ alternative on the basis that these individuals are more likely to have a pre-existing relationship with their doctor.5 However, as we found no significant variation to report, we use the results and findings from the basic main effects regression, Model 1. It was expected that patients with experience of consulting a NIP previously would be more likely to choose the ‘nurse prescribing’ alternative given the importance of experience has previously been demonstrated.11,12,15 Model 2 estimates the utility function of the subgroup of
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Who should be prioritized for renal transplantation? : analysis of key stakeholder preferences using discrete choice experiments

Who should be prioritized for renal transplantation? : analysis of key stakeholder preferences using discrete choice experiments

In this analysis we have used Discrete Choice Experi- ments (DCEs) in order to establish respondent’s valuation of different kidney transplant allocation criteria, and how they might trade-off gains in relation to one transplant al- location criterion, for losses in relation to another trans- plant allocation criterion. DCEs involve the application of a stated preference technique in order to establish a respondent’s valuation of attributes or characteristics of a good or service or health state. DCEs are increasingly being used to address priority setting issues in healthcare, both in primary care [12], and secondary care [13,14]. Some DCE research has been published on general trans- plantation issues, including assessing factors influencing willingness to donate body parts [15] and a DCE to estab- lish UK priorities for liver transplantation [16,17]. In renal transplantation, the first DCE findings internationally ema- nated from our study conducted in the UK [18]. This pub- lication focused solely on assessing whether patient preferences varied by ethnicity and gender. More recently, DCE research has been undertaken in Canada relating to patient and healthcare professional preferences for chronic kidney disease (CKD) care more generally (although not specifically focused on kidney transplantation) [19].
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Patients’ valuation of the prescribing nurse in primary care: a discrete choice experiment

Patients’ valuation of the prescribing nurse in primary care: a discrete choice experiment

Although field sites involved in the study were spread across England and delivered compara- ble services 20 the overall representativeness of the sample remains unknown. Any possible issue in the representativeness of the sample could have an impact on policy analysis and the use of findings to support any policy change. For this reason, it is important that future research pays close attention to understanding the limits of generalizing results, particularly in areas where there is greater social disadvantage. The DCE approach for valuing health care has become widely used in recent years. While it is true that much of the evidence gathered in this way has been shown to be reliable and internally valid more ought to be researched into demonstrating the external validity of results. External validity is, however, challeng- ing for any value-based measure applied to pub- licly funded health care services given the lack of a market. Other aspects, such as possible con- cerns about the appropriateness of the health care received might be important to patients when choosing between different health-care packages, although for our study the choice of these specific DCE attributes was supported by evidence from the literature, discussion with experts and pilot work with patients. Unfortu- nately, no costing data on the delivery of the alternative services were considered. Future work should integrate costing and DCE output within a cost-effectiveness framework to investi- gate how preferences (and their heterogeneity) might influence cost-effective decisions.
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Using discrete choice experiments to inform the design of complex interventions

Using discrete choice experiments to inform the design of complex interventions

This paper presents how DCEs can complement conven- tional qualitative research to prioritize service compo- nents most critical to the target population of adult men in Tanzania. The DCE results led to the inclusion of two key service components. Firstly, the strong preferences among both the younger men and the older men revealed the importance of age-separated waiting areas at the VMMC facilities. Waiting areas for VMMC were rearranged to allow for younger and older men to wait separately and out of view of each other. Secondly, the strong preference for partner counselling led to the introduction of partner counselling booths, staffed by both female health workers and female community peers who provided one-on-one and small group information for female partners about circumcision care and post-circumcision abstinence. Out-of-hours service provision was preferred in Njombe, but the utility was relatively small compared with the value placed on sepa- rated waiting areas and partner counselling service. This allowed the intervention to focus less effort on promot- ing out-of-hours VMMC services, although they remained available upon request. The suggested prefer- ence for opt-out testing led to an intensification of the information campaign, ensuring clients were comfort- able requesting not to be tested for HIV and staff were re-trained to emphasize the right to opt-out of HIV test- ing to clients. Officially, however, both the intervention and the control arms maintained a policy of opt-out HIV testing.
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Factors influencing job preferences of health workers providing obstetric care: results from discrete choice experiments in Malawi, Mozambique and Tanzania

Factors influencing job preferences of health workers providing obstetric care: results from discrete choice experiments in Malawi, Mozambique and Tanzania

The primary target for the DCE was health care workers who had performed at least one of the EmOC signal functions in the previous three months; thus the focus was on maternity staff, as well as health care workers who provide surgical services, such as caesarean section. Since it was not possible to randomly sample healthcare workers themselves, guided by existing staffing levels, the project randomly sampled hospitals and health cen- tres to be visited to approach the minimum target of 500 health care workers per country for the provider survey. Hospitals were intentionally oversampled because the majority of EmOC is provided in hospitals rather than health centres. In Malawi, a near-national sample of facilities (N = 84) intended to provide EmOC services was identified and included central, district, rural and CHAM (faith-based organisations) -operated hospitals and a randomly sampled urban and recently upgraded health centre designated to provide EmOC. A few districts/facilities were excluded in Malawi due to their recent participation in another human resources study in which similar data had been collected from health workers. In Tanzania, due to the size of the coun- try, cluster sampling was employed. One region was randomly selected in each of the eight geographic zones and all districts within those eight regions were then included in the sampling frame. The primary hospital serving the district was identified for inclusion; either the government-run district hospital or voluntary agency-run (VA) designated district hospital (DDH). In some districts that also contain the regional headquar- ters, the regional hospital was included in the sample when there was no district hospital serving the commu- nity. One health centre (HC) was randomly selected in each district, thus there were two facilities from each district in the study (N = 90). In Mozambique, a near national sample of general, district and rural hospitals was included to maximise the potential participation of the NPC cadre tecnico de cirurgia. In addition, two to three health centres (type 1 and type 2) providing maternity care, and therefore at least some basic EmOC functions, were Table 2 Attributes and attribute levels for job alternatives –
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Patients' preferences for GP consultation for perceived cancer risk in primary care: a discrete choice experiment.

Patients' preferences for GP consultation for perceived cancer risk in primary care: a discrete choice experiment.

Prompt consultation in primary care when people experience a potential cancer symptom is considered key to earlier diagnosis. Several factors are known to influence consulting, however, it is not clear how the public weigh up these factors. This study used a discrete choice experiment to explore preferences for GP consultation using different cancer symptom scenarios. Listening skills of the doctor was a key driver of preference; improving communication within consultations is likely to encourage help-seeking longer-term.

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Women’s preferences for maternal healthcare services in Bangladesh: evidence from a discrete choice experiment

Women’s preferences for maternal healthcare services in Bangladesh: evidence from a discrete choice experiment

Maternal mortality in Bangladesh impacts inequities in access to primary health services with a huge gap between women in advantaged and disadvantaged socioeconomic communities [1]. About 800 women die per day due to preventable causes associated with reproductive health concerns [2]. Furthermore, the lifetime threat of maternal deaths related to childbearing was higher in remote areas and low resources settings [3]. According to the World Health Organization (WHO), about 99% of maternal deaths happen in low- and middle-income countries [4]. In 2015, the maternal mortality ratio (MMR) was 239 per 100,000 live births in low- and middle-income countries compared to 12 per 100,000 live births in high-income countries [4]. Bangladesh has in recent decades made significant achievements in maternal health indicators [5], with MMR being dramatically reduced [6–8]. Several initiatives undertaken by the government of Bangladesh includes introducing appropriate preventive strategies and effective collaboration with non-government organisations and other stakeholders. A number of initiatives such as emergency obstetrical care services and maternal health voucher schemes contributed significantly to improving primary healthcare outcomes [9]. However, this overall reduction does not accurately outline the whole picture of maternal healthcare status in the country [10]. Despite an increase in demand for maternal healthcare, limited supply and difficulties in access during pregnancy and delivery still leave a large number of women at higher risk for preventable death [11]. The individual demand for accessing quality maternal healthcare increases the maternal and child survival rates and significantly affects the mortality transition [5,12].
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Inferring Attribute Non-attendance from Discrete Choice Experiments: Implications for Benefit Transfer

Inferring Attribute Non-attendance from Discrete Choice Experiments: Implications for Benefit Transfer

While it is implicit that the parameters are constrained to be equal across classes in the ECLC, it is less obvious if this should also apply to ASCs. Previous literature applying the ECLC is mostly silent about the treatment of ASCs, although it can impact on results. An exception is Hensher et al. (2012), which estimates ASCs for the one class where all attributes are attended to, but omit constants (constraining ASC coefficients to zero) for all other classes where one or more attribute coefficients are assigned zero values. In environmental DCE applications, an ASC is usually added to capture differences in utility between the status quo alternative (or an ‘opt-out’ alternative) and the other choice options. It may be possible to assign a behavioural interpretation to the status quo option, for example related to protest attitudes (Meyerhoff and Liebe 2009), if the attribute levels used to describe the status quo also appear in the designed part of the choice experiment. If this is not the case, however, it is not possible to estimate the difference in utility between the baseline level of the attributes (used in the status quo option) and the reference level (the lowest level appearing in the designed part of the choice experiment). In this situation, the ASC parameter captures the differences in utility between the baseline level and the reference levels for all the attributes, together with any utility from choosing the status quo that is unrelated to attribute information. In an ECLC context, this means that it is counterintuitive to constrain the ASC to be equal across classes, as non-attendance to one or more of the attributes will almost certainly influence the ASC parameter. Constraining the ASC to be equal hence imposes unnecessary restrictions and may bias inferred AN-A patterns, preference estimates and MWTP estimates. Therefore, we do not impose ASC equality across classes in our case.
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<p>Preferences for the administration of testosterone gel: evidence from a discrete choice experiment</p>

<p>Preferences for the administration of testosterone gel: evidence from a discrete choice experiment</p>

Symptoms associated with the condition include sexual problems such as erectile dysfunction, delayed ejaculation, reduced libido and lowered fertility, as well as decreased muscular formation, cognitive dysfunction, depression, decreased erythropoiesis (red blood cell production and visceral obesity). 2,5 These low male sex hormone levels are, in particular, associated with metabolic and cardiovas- cular diseases such as hypertension, mild dyslipidemia, insulin resistance, Type 2 diabetes, and atherosclerosis. 5–7 Treatment of TDS most commonly involves testoster- one replacement therapy (TRT) and is usually considered for patients who are persistently suffering from erectile dysfunction and/or diminished libido and low testosterone levels. 1,4 TRT aims to increase testosterone levels in order to minimize the symptoms associated with TDS. 8 TRT can be administered via tablets, patches, implants, injections, or gels. Different modes of administration are associated with differences in application site and frequency of administration, as well as the time taken to achieve and maintain serum testosterone levels as the desired therapeu- tic effect. 9
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Identification of Time Preferences in Dynamic Discrete Choice Models: Exploiting Choice Restrictions

Identification of Time Preferences in Dynamic Discrete Choice Models: Exploiting Choice Restrictions

time. Furthermore, the last period for which differences exist and the period after that have to be observed. Time preferences can then be identified from these two periods in the same fashion as in section 3. In contrast, time preferences are not necessarily point identified if researchers observe the data for two groups of economic agents that exclusively differ in their restriction probabilities. To show this, let δ ∈ {A, B} denote a group indicator. Restriction probabilities depend in this case not only on state-choice combinations but also on the group indicator. I denote these by π (d, x, δ). Because of the different restriction probabilities, value functions differ between the two groups. I denote the vectors m ( ˆD,d) and v (d) by m( ˆD,d,δ) and v (d,δ) for this demonstration, where ˆ D ∈ {D, ̃ D (d); d ∈ D}. Similarly, I denote the genuine choice probabilities by GPr (d ∣ ˆD,x,δ). In this case, (6) can be written as
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