Minimum Wage Effects on Job Attachment: AGender Perspective

We examine whether the effects of the introduction of a minimumwage on low-pay employment duration in Germany in 2015 are heterogeneous by gender. In order to disentangle the effects on women and men, we estimate a duration model with unobserved heterogeneity in which we allow gender differences and differences before and after the introduction of the minimum wage. We find that the reform does affect women and men differently, in particular, it mainly increasesmen’s job attachment. These gender differences in job attachment are the strongest for full-time employment. In consequence, although the minimum wage may have been set up as a gender-neutral instrument, in an indirect way, it affects women and men differently. We discuss different mechanisms that could account for our result and carry out several robustness checks.

high for female workers than for male workers, where low pay is defined as less than two-thirds of the national median or the national mean of gross hourly wages (Schnabel 2016). For Germany, this is also true. Data from the Federal Statistical Office for Germany shows that 26.4 % of employed women are on low pay employment, while for men this number only is 16.3 % (Federal Statistical Office 2018). This is partially a consequence of almost 50 % of employed women being working part-time and 66 % of workers with mini-jobs being women (Federal Employment Agency 2019).
The fact that women are over-represented in low paying jobs is a factor contributing to the observed gender wage gap as well as having consequences for life-time income, leading to lower pensions and differences in poverty status. In particular, women have on average half the amount of pension than men have, making them more likely than men to be at risk of being poor when old. Indeed, in 2018, 20 % of over 65 year old women were at risk of being poor, while this percentage was 17 % for men (Federal Statistical Office 2019). Therefore, understanding whether a minimum wage policy can help to reduce gender differences is a relevant issue from a public policy perspective.
The standard model of competitive labor markets predicts that the introduction of a minimum wage that is set higher than the competitive equilibrium wage reduces employment among low-skilled workers. However, empirical research has disputed this theory. For the US, the seminal paper by Card and Krueger (1994) found no effects on the employment in the US fast food industry after the 1992 increase in New Jersey minimum wage. Since then, other studies have found no effects or small effects going in both directions. A meta-analysis of 23 minimum wage studies by Belman and Wolfson (2014) finds that these studies were as likely to find positive as negative effects of the minimum wage and the estimate typically was close to zero. For the UK, Stewart (2004) finds that the introduction of the national minimum wage in April 1999 did not harm the employment prospects of the low-paid. However, Dickens et al. (2015) find that the introduction of a minimum wage reduces employment retention of part-time working women. They argue that Stewart (2004) finds no effect as he did not consider the effects on the part-time employment of women who are one of the most vulnerable group to changes in the minimum wage.
Thus, the results in the literature of no effects on employment might hide heterogeneous effects by gender.
There are several empirical studies of the introduction of the minimum wage in 2015 in Germany. This minimum wage of 8.50 euro considerably exceeds the average wage of the affected workers before the reform of 6.01 euro (Bossler and Schank 2017) and, thus potentially matters for labor market outcomes. These studies exploit variation in the minimum-wage at the regional or establishmentlevel for treatment assignment in a diff-in-diff setting. In particular, Caliendo et al. (2018) and Bossler and Gerner (2016) find that in the short run the introduction of the minimum wage had a moderate negative effect on overall employment (see Caliendo et al. 2019 for a literature survey on the causal effects of the minimum wage introduction in Germany). Garloff (2019) and Schmitz (2019) also find small negative (or insignificant) effects on overall employment driven mainly by less marginal employment and Holtemöller and Pohle (2020) find negative effects on marginal employment while finding small positive effects on regular employment. 1 As for the effects on wages and the gender wage gap, Boll et al. (2015) obtain via simulation studies that the minimum wage increases women's wages and thus decreases the gender gap. Bossler and Schank (2017), on the other hand, estimate that the introduction of the minimum wage can account for about half of the recent decrease in wage inequality. They find a positive wage effect on all affected employees (up to the median), but also a positive wage effect on the existing jobs. Finally, Caliendo and Wittbrodt (2020) find that the minimum wage reduced gender wage disparities, in particular, the regional gender wage gap.
We take another approach by analyzing gender differences in the 2015 minimum wage reform's effect on low-wage workers' employment duration. In particular, we use data for Germany for the period 2013-2016 and estimate a duration model with unobserved heterogeneity. We allow for gender differences and differences before and after the introduction of the minimum wage.
The most related paper to ours is Phimister and Theodossiou (2009). They examine gender differences in the duration of low pay employment spells prior to and after the introduction of a national minimum wage in 1999 in the UK. Specifically, they consider how gender differentials in the duration of low-paid jobs and the probability of progressing to a higher-paid job change due to the introduction of a minimum wage. While they investigate gender differences by estimating separate specifications for men and women, we carry out a joint estimation which allows us to jointly test for gender differences and for reform effects. While separate estimations are mainly a heuristic device, our joint estimation allows to properly disentangle effects from gender and from the reform. In particular, it enables us to statistically test whether men and women's job attachment differs before and after the reform. In effect, we find that the reform affects mainly men's separations. This leads the job separation rates of women and men to converge.

Methodology
We exploit the quasi experimental nature of an introduction of a minimum wage in Germany at the federal level in 2015. In particular, we analyze the duration of low pay employment for men and women, as well as the probability of becoming unemployed or inactive. First, we estimate a single spell discrete time hazard model with one exit type to unemployment or inactivity. At a second stage, unobserved heterogeneity is incorporated by assuming a proportional hazard model with Gaussian Mixing (Andrews et al. 2002;Lancaster 1990).
Each person in low pay employment potentially exits low pay in an interval j − 1, j [ ]to one state. The latent variable, T un represents the potential time in low pay with exit of type un. The random variable T = min T un ( ) represents the low pay duration. The hazard h j corresponds to the probability of an exit to state un during period j given that the low pay duration lasted to j − 1. Finally, the survival function is . Specifically, we model the exit type using a proportional hazard model with Gaussian mixing, i.e.
where h p, q j0 , p ∈ m, f { }; q ∈ u, t { } are baseline hazards for any combination of men (p = m) and women (p = f) as well as untreated (q = u) and treated (q = t) (i.e. before and after the introduction of the minimum wage). v i is a random variable capturing unobserved heterogeneity so that u = log v ( ) is normally distributed. x i is vector of covariates including individual and job characteristics (specified below).
The probability of exiting employment is given by, the expected low pay duration given an exit, by and finally the expected low duration by ΠE. While Phimister and Theodossiou (2009) separately estimate the modelallowing for gender differencesbefore and after the introduction of the minimum wage, we jointly model both dimensions. Therefore, because of the two dummies -gender and introduction of minimum wageand their interactions both baseline hazards and the impact of each characteristic are allowed to vary across male and female, as well as across the samples before and after the introduction of the minimum wage. More importantly, this enables us to do inference along both dimensions rather than merely descriptively compare them. In particular, it enables us to statistically test whether there are gender differences in the job attachment and in the role of other covariates before and after the reform.

Data
We use data from the German Socio Economic Panel, SOEP(v34) (2019) for the years 2013-2016. We use spell monthly employment datadifferent from most previous studiesallowing us to have a more complete picture of how employment dynamics change due to the introduction of a minimum wage across gender. Ever since its start in 1984, the SOEP contained a calendar section asking about employment status as of January through December of the previous year. An employment status spell can take several values such as full-time employed, parttime employed, unemployed, housewife/househusband, short-work hours, maternity leave. We define employment in a broad sense if the spell type is either full (1), part (3), short-work hours (2) or mini-job (15). We define out of the labor force (unemployed or inactive) if the spell type is unemployed (5), housework (10) or other (12). Moreover, there are spells where individuals report to be in different statuses at the same time (overlapping spells). For example they report to be working part-time and for the same spell time they report being a housewife. We treat those spells as employed if one of the statuses reported corresponds to our definition of employment while the other spell is reported to be different to registered unemployed.
We define low-wage earners as those whose hourly wage belongs to the bottom 25th percentile of the hourly wage distribution for each year. Using relative measures is standard in the literature (see Phimister andTheodossiou 2009, Cappellari andJenkins 2008). To construct our threshold for low-wage earning we use contractual weekly working hours and gross monthly wages adjusted by average weeks per month to calculate the hourly wage.
From the constructed data set, two inflow samples of low-pay spells were drawn covering two two-year periodsprior to and post the introduction of the minimum wage. Specifically, low pay spells starting between January 2013 and December 2014 are included in the pre-minimum wage sample, and spells starting between January 2015 and December 2016 are included in the post-minimum wage Minimum Wage and Job Attachment sample. All continuing spells are treated as censored if they continued beyond the end cut-off points, i.e. December 2014 and December 2016 respectively.

Covariates
Various variables at the household and individual levelsuch as sector, work experience, firm tenure, education, age, marital statusare included as covariates. Table 1 provides descriptive statistics for our variables of interest. Previous research shows that not only gender but also human capital, job tenure and experience and part-time working, are all important in low pay mobility (Phimister and Theodossiou 2009).
In our sample, women account for more than 60 % of low paid jobs. We can observe that women and men are similar before and after the reform in most covariates. Men have slightly less education (less with secondary education) and are less likely married than women. A stark difference is the share of individuals who are in a full-time employment or in a mini-job. The share of men in a full-time low paid job is double the share of women in full-time while the opposite is true for mini-jobs. The average hourly wage in the data-set was, before the introduction of the minimum wage, 6.86 euro for women and 6.88 euro for men, and after, 7.57 euro for women and 7.27 euro for men. This reflects the wage increase due to the introduction of the minimum wage. Surprisingly, after the reform there is an unconditional negative gender wage gap, i.e. in this wage segment, the average wage of women is larger than the one of men.

Association Analysis Between Gender, Job Separation and Reform
We use contingency tables to analyze the association between job separations and reform for women and for men. Table 2 shows the absolute frequency together with the proportion conditional on the reform status. However, the reduction is larger for men (from 1.32 % to 0.87 %) than for women (from 1.01 % to 0.88 %), suggesting that job separation rates of women and men might converge after the reform. Pearson χ 2 1 tests confirm that while for women there is no association between reform and separations (test statistic of 1.81 with a p-value of 0.178), for men there is one (test statistic of 10.17 with a p-value of 0.00).  Figure 1 displays the Kaplan-Meier unconditional survival functions by gender and reform status. The survival function estimates the probability of a low-paid job to survive period j conditional on having survived until then. Thus, we can observe from the figure that the survival of men after the reform (the line "male after") is the highest. Thus, the probability of remaining in a low paid job is highest for men who enter a job in 2015-2016. Compared to the survival function before the reform for men ("male before"), they seem to be different. It is less clear if women's survival functions are different before ("female before") and after ("female after") the reform and whether after the reform, men's and women's survival functions are different. We can test formally whether these survivals are different by carrying out log-rank tests for equality. We do pairwise comparisons of the survival functions. The null hypothesis is that the survival functions are equal. For the prereform period, 2013-2014, we reject the equality of the survival functions for women and men (p-value = 0.00). However, for the post-reform period, 2015-2016, the equality of the survival functions for women and men cannot be rejected (p-value = 0.98). Before the reform, men were less likely to remain in a job than women, while after the reform, there are no differences in the unconditional survival of a job. On the other hand, the unconditional survivals are equal before and after the reform for women (p-value = 0.30), while for men we reject the equality of their survival before and after the reform at the 1 % level (p-value = 0.00). Thus, the unconditional survival functions might suggest that men starting a job in 2015-2016 are less likely to become unemployed or inactive than those men starting work in 2013-2014, while women's survival is not affected by the introduction of a minimum wage. Therefore, the reform appears to have a positive job attachment effect on men lifting their previously lower attachment level to the one of women. However, these are unconditional survivals and differences might be due to other factors in which women and men differ. Therefore, we next have to control for characteristics that alternatively might explain these results. In particular, we saw before that there are stark differences in how much time men and women spend at work, men participating more full-time than women and women being more likely to have a mini-job. Therefore, we pay extra attention to these variables in the estimations that we carry out in the next section.

Econometric Results
We estimate a single spell discrete time hazard model with one exit type to unemployment or inactivity. The dependent variable is then "exited employment at month t to unemployment/inactivity". We control for reform (reform = 1 if year is 2015 or 2016, post reform), gender (female = 1 if woman), as well as a baseline hazard modeled as a step function with different step-sizes, the first year is divided into three steps, two-three months steps at the beginning, followed by a six-months step while the second year is considered as one step. The steps are captured by the dummy variables d456 for the months 4-6, d712 for the months 7-12 and d1324 for the months 13-24. We take this approach to keep the number of steps low and to represent higher action in exits in the first months of employment. We further control for the set of variables described above. Moreover, given that we estimate our model jointly for women and men and before and after the reform, we are able to control for "triple" interactions of several employment-related covariates in which men and women exhibited stark differences before and after the reform, such as fulltime employment (fulltime), years of experience in full-time employment (expft) and having a mini-job (minijob), as well as the step function with gender and reform. In a first step, in Subsection 4.1, we do not include unobserved heterogeneity but we do so in a second step in Subsection 4.2 and test for whether it matters.

Without Unobserved Heterogeneity
First, we assume there is no unobserved heterogeneity in equation (1), i.e. v i = 1. Differences between observations are then explained by observed characteristics or Minimum Wage and Job Attachment are purely random. The more general case of unobserved heterogeneity is considered in Subsection 4.2. Table 3 reports the results of our estimations. In particular, we examine whether there are gender differences in the reform's effect on employment duration, and, whether there are gender differences in the effect of other covariates. We start in column one with a simple model where exit from employment is potentially affected only by gender, the reform and the baseline hazard. In the second column, we include also all controls. In the third column, we report the full model in which we control for all covariates, the interaction of several covariates with gender, with reform and with both gender and reform (triple interactions) as well as the interactions of the steps of the hazard function with gender (femalehazard), reform (reform-hazard) and gender and reform (reform-female-hazard). 2 These interactions are of special importance for understanding gender differences in the reform's impact on the effect of these variables on employment duration. Comparing column 1 to column 2, we can observe that the coefficients remain constant and significant once we control for covariates. In particular, the coefficients of female and reform as well as the hazard function are significant in both specifications. Moreover, having a temporary contract, working in manufacturing relative to working in the service sector and working in a firm with less than 20 employees have a significant effect on employment duration. Column 3 presents a full model in which we allow for all possible interactions between reform and gender and the hazard function, full-time employment, having a mini-job and a measure of work experience (work experience in full-time work). Once we include interactions of the hazard function and employment-related covariates with reform and gender in column 3, the full-time interacted terms capture the previous significance of the reform and gender (column 2). In consequence, differences of the reform's effect between women and men are especially relevant for those that work full-time.
To arrive at our final model in column 4, 3 we remove the triple interactions that were insignificant in the full model (column 3) after testing formally for joint significance. In particular, these are triple interactions of reform and gender and the hazard function (reform-female-hazard which summarizes the triple interaction with the three steps of the hazard function, reform-female-d456, reform-female-d712, reform-female-d1324), having a mini-job (minijob-reform-female) and work experience (expft-reform-female). The F-test statistic for the joint null hypothesis of the coefficients of (i) reform-female-hazard, (ii) minijob-reform-female and (iii) expftreform-female being equal to 0 is 4.19 with a p-value of 0.52. This allows to make the model more comprehensible and to facilitate the interpretation.

Minimum Wage and Job Attachment
We cannot reject the hypothesis that these interactions are jointly insignificant, therefore, we do not include them in our final model in column 4. The significant coefficients remain the same as in the full specification, the only difference being that the last step of our hazard function becomes significant. The interaction of full-time with gender (fulltime-female), with reform (fulltime-reform) and with gender and reform (fulltime-reform-female) are all significant. We can interpret the coefficient of fulltime-female as capturing the effect of being a woman in full-time employment on the probability of exiting employment before the reform. In particular, we find that before the reform, full-time is more strongly associated with job duration for women than it is for men. We can interpret the negative coefficient of fulltime-reform as a larger employment duration effect from the reform for men in full-time than for men not in full-time. Finally, the positive coefficient of the triple interaction fulltime-reformfemale of 1.006 measures the diff-in-diff reform effect from full-time versus part-time for women relative to the one for men. In consequence, after the reform, men benefit more from a full-time employment (in terms of job duration) than women do.
Next, we carry out F-tests of joint significance to establish whether the introduction of the minimum wage, gender and an interaction of both affect employment duration in our final model. We report three different tests in Table 4. Our first test is whether gender has an effect on employment duration. In particular, the hypothesis we test is whether the coefficients of female, female-hazard, fulltime-female, minijobfemale, expft-female and fulltime-reform-female are jointly equal to zero. We reject that gender does not affect exit into unemployment or inactivity at the 10 % level (pvalue = 0.085). In the second column, we consider whether the reform affected employment duration. Here, the hypothesis we test is whether reform, reform-hazard, fulltime-reform, minijob-reform, expft-reform and fulltime-reform-female are jointly equal to zero. Once again, we reject the hypothesis of no effect at the 10 % level (p-value of 0.071), and we find that the introduction of the minimum wage affected employment duration. Our final and most interesting test is whether women's employment duration was affected differently by the reform than that of men (where the null hypothesis is reform-female and fulltime-reform-female are jointly equal to zero), for which we reject the hypothesis of no effect at the 5 % level (p-value of 0.032). Therefore, the results support our hypothesis of gender differences in the reform's effect on employment duration. Among others, through the reform, men benefit more from a full-time employment (in terms of job duration) than women do (as seen in the coefficient of the interaction of female and reform with full-time, 1.006). A potential explanation for this finding could be that men value wages more while women prefer different job features instead (see the discussion together with empirical evidence in Section 5).

Frailty
Up to now, we have not allowed for unobserved heterogeneity. By taking into account unobserved heterogeneity, we let unobserved characteristics at the individual level affect employment duration. We estimate a frailty model with v i in equation (1) following a Normal distribution (presented in Table 5 4 ). The LR test of absence of unobserved heterogeneity is significant at the 1 % level (see second to last row in Table 5), so we should indeed allow for heterogeneity. Column 3 reports the final model, where most of the variables that were significant under no frailty remain so. 5 Also with frailty, we find that after the reform, men benefit more from a full-time employment (in terms of job duration) than women do (significant positive coefficient of the triple interaction fulltime-reform-female of 1.175). We test again for our three hypotheses in Table 6 and find that gender does not affect employment duration once we allow for unobserved heterogeneity (with a p-value slightly above 0.1). However, the reform has still an effect on employment duration (at the 10 %), and, most importantly, women's and men's employment durations are affected differently by the introduction of the minimum wage (at the 5 % significance level). Therefore, we find that our results remain largely unaffected when allowing for unobserved heterogeneity.

Discussion of Results
The main result we find with the different specifications, both without and with unobserved heterogeneity, is that the reform affected men and women differently.  In particular, the exit rate of men decreases while the one of women remains constant.
First we consider a supply side mechanism where workers react to the introduction of a minimum wage. Recall that the introduction of the minimum wage results in a positive wage effect (Bossler and Schank 2017). We would expect that an increase in wages would lead to a higher job value for workers so that they would be more likely to choose to keep their jobs. While we do observe this for men, this is not the case for women. A possible reason is that if women valued not only wages in a job, an increase in wages would not necessarily translate into a higher value of a job. Indeed, there is evidence that women value wages less than men do and prefer different attributes, see Fortin (2005), Blau and Kahn (2017). Therefore, women might trade in lower wages for some other attribute, such as flexibility. Indeed, Goldin (2014) finds that preference for flexibility has a negative effect on wages. Thus, if women in low-wage jobs valued flexibility, an increase in wages created by the introduction of a minimum wage might not necessarily increase the value of their job and consequently might not lead to higher attachment. Relatedly, this could imply that for men the substitution effect is dominant, while for women it is not, insofar as the increased wage makes working relatively more attractive to men, but not women.
In principle, there could also be a demand side mechanism which is however not supported by the data. Firms might value a job match with men higher than with women as they expect women to have more career interruptions. Before the reform, firms could then offer lower wages to women to account for the expectation of more interruptions. With a minimum wage, there would be restrictions on these, and firms then would maximize the expected value of the match by hiring men. Firms could not adjust for women's preference of flexibility with lower wages, thus, they would be more likely to break the match.
Both the association analysis and the unconditional survival functions suggest that the women's likelihood of job separation remains similar after the reform. Therefore, we argue that the supply side explanation might be more relevant for our results since we would expect firms to increase separations of women if it was mainly demand side driven.

Robustness Checks
We carry out two robustness checks to confirm our results: first, we allow for a more flexible baseline hazard and, second, we consider actual hours worked instead of the dummy for full-time employment. In Table 7 we report our results of the robustness checks. In the first column, we allow for a more flexible baseline hazard and, in the second column, we consider actual hours worked instead of a full-time dummy. We only report the variables of interest we emphasized in the main analysis (see the detailed estimation in Tables A.10 and A.11 in the Appendix A). We rename the variable "full-time" to "work time" to refer to both of the two different variables: a dummy that takes the value of one if the worker works more than 30 h per week in (1) and actual hours per week in (2). We start by allowing for a more flexible baseline hazard. In our final estimation, column (3) in Table 5 the first year is divided into three steps, two three-months steps at the beginning, followed by a six-months step while the second year is considered as one step. This baseline hazard choice could be too broad, so we allow for a baseline hazard with three-months steps over the whole two year period and estimate our final model with this alternative baseline hazard.
In the first column of Table 7, we can observe that allowing for a more flexible baseline hazard does not affect our results, we still find that the reform had different effects on women and men (F-test for joint significance of the triple interactions is 6.47 with p-value = 0.039) and the size of the effect remains stable, when compared to Table 5. In the last column, we change the definition of full-time. We control for actual hours worked per week instead of the dummy for full-time employment. Thus we provide a more flexible variable that can control for time at work. We find again, that our results of the reform having different effects by gender hold (F-test for significance of the triple interaction is 10.36 with p-value = 0.0056). As for the coefficient of the triple interactions, work time-female-reform, both coefficients (1.141 for the specification with the more general hazard and 0.049 for the specification with the actual work hours) are positive and significant. Note that for the actual hours, the scale is different, and thus the coefficients differ as well. The discovered gender difference is then even more general, insofar as not only full-time working but even each additional worked hour increases this difference.

Conclusions
Women's economic labor market outcomes still lag behind the ones of men. Several gender differences are well documented. Among others, women are overrepresented in the low wage sector.
Using data for Germany, we examine whether the effects of the introduction of a minimum wage on low-pay employment duration in 2015 differ by gender. To do so,  we estimate a joint duration model for men and women before and after the reform. We show that a minimum wage aiming at improving the situation for low wage workers affects women's and men's employment duration differently. In particular, we find that the reform had an effect on employment duration and that this effect is higher for men than for women. These gender differences in employment attachment are the strongest for full-time employment. This presents an important outcome since an increase in persistence in full-time employment seems more desirable than one in part-time or in mini-jobs. We argue that men's higher job attachment could be in line with a supply side mechanism according to which women, unlike men, might place a higher value on other job characteristics such as flexibility rather than wages. Therefore, an increase in wages due to the introduction of a minimum wage would not increase the value of a job for women but would increase the one for men. This would lead to an observed higher job attachment for men than for women after the reform. This dominance of the supply side effect (over an alternative demand side effect) would be in line with Bossler and Gerner (2016) and Bellmann et al. (2016) who suggest that labor demand is mainly adjusted through a reduction in hiring (which would not be reflected in job separations) rather than firing. In any case, a more thorough analysis to disentangle supply from demand side effects could be interesting. For doing so, we would need to separate employee-initiated and employer-initiated job separations.
Unfortunately, the variable inquiring the type of job separation in our dataset has too many missing values, making a meaningful analysis unattainable. This distinction would be further relevant for concluding whether the resulting higher job attachment as consequence of the minimum wage reform indeed is a desirable feature. In particular, for employer-initiated job separations, a higher attachment should be desirable. Employee-initiated job separations, on the other hand, are not necessarily undesirable since a certain amount of job mobility might help to climb the job ladder to eventually escape the low wage segment of the labor market.
Our results highlight the importance of understanding that reforms have heterogeneous effects, women's and men's employment durations are affected differently by an introduction of a minimum wage. In particular, while men's job duration increases, the one of women remains unchanged. The minimum wage could then have misleading effects on the gender wage gap insofar as it (partially) decreases due to a selection effect in which men increasingly remain in the lowwage sector. It could be interesting to quantify the effect on the gender wage gap in view of Caliendo and Wittbrodt (2020) finding of a decreased gender wage gap. In any case, Bossler and Schank (2017) aforementioned finding of a positive wage     Source: SOEP,  to . The dependent variable is "exited employment at month t". The coefficients are marked with a if the level of significance is between  % and  %, b if the level of significance is between  % and  % and c if the level of significance is less than  %. LR test (frailty) is a test for frailty (null hypothesis of no frailty) together with the p-value.  , Source: SOEP,  to . The dependent variable is "exited employment at month t". The coefficients are marked with a if they are significant at the  % level, b if they are significant at the  % and c if they are significant at the  % level. LR test (frailty) is a test for frailty (null hypothesis of no frailty) together with the p-value.