help with stcurve after cox model

[Cheng Li]

Hi,
I hope to get some help on interaction term plot and interpretation. I have run a Cox hazard proportional model and plotted two significant interaction terms by holding all at mean value except the highlighted variable (one sd above and below). You can also see the interaction term plotting below. I used stcurve to plot both. My question is why the graph appears to be polynomial. I thought it'd be a smooth line as everything is linear after I take two values from the interaction term and split the term into two.

H4a: The more international a firm is, the weaker the effect of home country collective bargaining on MNEs’ commitment to a GFA.
H5b: The more profitable a firm is, the stronger the effect of home country individual rights on MNEs’ commitment to a GFA.

Thanks,
Cheng

Dependent variable = Duration of adoption		1	2	3	4	5	6	7	8	9
Collective bargaining legitimacy	H1		1.016**			1.681***		1.008		1.759***
			(0.00745)			(0.300)		(0.00936)		(0.323)
Individual employee rights	H2			2.477***			0.178		1.433	1.702*
				(0.456)			(0.510)		(0.332)	(0.467)
Collective bargaining legitimacy * Internalization	H4a					0.601***				0.568***
						(0.108)				(0.105)
Individual employee rights * Internationalization	H4b						13.54
							(39.46)
Collective bargaining legitimacy * Profitability	H5a							1.001
								(0.00116)
Individual employee rights * Profitability	H5b								1.093**	1.085**
									(0.0413)	(0.0441)
Internationalization						3.617e+07**	0.490			9.196e+08**
						(2.562e+08)	(3.751)			(7.468e+09)
Profitability								0.933**	0.767***	0.780**
								(0.0308)	(0.0696)	(0.0764)
Mining					11.57***	9.780***	4.450*	8.685***	5.150*	4.221*
					(9.416)	(8.041)	(3.794)	(7.167)	(4.417)	(3.685)
Construction	H3				28.46***	24.73***	11.17***	17.30***	9.780**	9.200**
					(24.60)	(21.52)	(10.16)	(15.41)	(9.009)	(8.509)
Manufacturing					5.777**	4.923**	3.049	3.959*	3.143	2.649
					(4.125)	(3.541)	(2.281)	(2.886)	(2.356)	(2.024)
Transportation					6.197**	4.910**	2.935	3.780*	3.070	2.403
					(4.773)	(3.876)	(2.405)	(3.021)	(2.512)	(2.035)
Finance					6.706*	7.546*	4.557	4.866	3.332	4.221
					(7.561)	(8.565)	(5.305)	(5.545)	(3.894)	(5.050)
Service					2.519	2.069	1.548	1.805	1.787	1.558
					(2.058)	(1.736)	(1.322)	(1.501)	(1.521)	(1.361)
Firm age		1.001	1.000	1.000	0.999	0.998	0.998	0.999	0.998	0.998
		(0.00222)	(0.00240)	(0.00218)	(0.00241)	(0.00254)	(0.00231)	(0.00246)	(0.00225)	(0.00234)
GDP per capita		0.527	0.408	1.648	0.799	0.511	1.349	0.662	1.954	1.501
		(0.289)	(0.226)	(0.862)	(0.444)	(0.305)	(0.757)	(0.393)	(1.129)	(0.935)
GDP per capita growth		0.866	0.860	0.891	0.885	0.867	0.885	0.898	0.934	0.912
		(0.0853)	(0.0823)	(0.0928)	(0.0943)	(0.0908)	(0.0957)	(0.0934)	(0.101)	(0.0981)
Firm size		1.166	1.204	1.212	1.433***	1.449***	1.343**	1.474***	1.419**	1.403**
		(0.133)	(0.139)	(0.151)	(0.193)	(0.197)	(0.193)	(0.203)	(0.205)	(0.210)
UN global compact		1.159	0.991	0.939	1.755*	1.408	1.149	1.451	1.254	1.116
		(0.347)	(0.302)	(0.277)	(0.568)	(0.487)	(0.377)	(0.502)	(0.413)	(0.385)
Listed		0.439**	0.326***	0.564*	0.237***	0.220***	0.412**	0.253***	0.369**	0.436*
		(0.149)	(0.120)	(0.196)	(0.0955)	(0.0937)	(0.178)	(0.113)	(0.157)	(0.207)
Solvency		0.994	0.992	0.995	1.002	1.001	1.003	1.009	1.009	1.011
		(0.00775)	(0.00801)	(0.00880)	(0.00963)	(0.00977)	(0.0102)	(0.0105)	(0.0113)	(0.0115)
High skill industry		1.317	1.250	1.244	2.143**	2.123**	1.806	1.974*	1.540	1.514
		(0.323)	(0.307)	(0.306)	(0.759)	(0.781)	(0.661)	(0.718)	(0.568)	(0.577)
Host country count		1.004	1.006	1.008	0.997	0.999	1.005	1.001	1.006	1.007
		(0.00600)	(0.00612)	(0.00616)	(0.00621)	(0.00649)	(0.00642)	(0.00656)	(0.00657)	(0.00670)
Sustainability report		7.816***	6.035***	5.500***	11.13***	10.10***	7.273***	8.722***	6.878***	7.035***
		(2.218)	(1.842)	(1.641)	(3.721)	(3.643)	(2.560)	(3.054)	(2.356)	(2.576)
LR chi2(22)		80.62	85.25	107.39	105.44	115.79	125.79	114.88	132.48	142.91
Log likelihood		-288.88985	-286.57441	-275.50271	-276.48132	-271.30618	-266.30572	-271.75963	-262.96131	-257.7454
No. of failures		71	71	71	71	71	71	71	71	71
No. of subjects		142	142	142	142	142	142	142	142	142
Number of obs		1799	1799	1799	1799	1799	1799	1799	1799	1799
Standard error in parentheses
In this table, all independent variables and control variables are lagged by one year.
*** p<0.01, ** p<0.05, * p<0.1

[Clyde Schechter]

I thought it'd be a smooth line as everything is linear after I take two values from the interaction term and split the term into two.

There is actually nothing even remotely linear in this context.

Let's review the Cox proportional hazards model. First there is some "baseline" survival function (probability that the duration of a GFA, whatever that is, is at least amount of time t.) By "baseline" we mean the survival function that describes duration of a GFA when all of the covariates in the model are zero. The Cox proportional hazards model says that when you have non-zero covariates, then the survival function changes, by applying a constant ratio to the hazard function of the baseline survival curve. Recall that the hazard is -d(ln S)/dt where S is the survival function and t is time. Moreover the particular ratio to be applied to derive the hazard function is exp(b0 + b1X1 + b2X2 +... + bnXn) where the X's refer to the values of the model covariates, and the b's are the coefficients that -stcox- estimates for you. Notice that we have layer upon layer of nonlinearity here already. The only thing that is linear is the b0 + b1X1 + b2X2 + ... + bnXn, but between the exponentiation of that to derive the hazard ratio, and the logarithmic derivative relating the hazard to the survival function, that linearity is completely drowned out. And I haven't even mentioned that the baseline Survival function itself is usually highly non-linear to start with.

Your plots are graphs of S versus t with the various covariates held at the values you stipulated in your -stcurve- command, and shown separately for different values of profitability, and internationalization. As you can see from the previous paragraph it would be miraculous if any of these graphs were even close to linear. So miraculous that I have never seen that happen, and if I ever did, my first reaction would be that there must be some pretty serious errors in the data or the coding.

In addition, I note that although your full model contains interaction terms involving profitability and internationalization, neither of these graphs could be called an interaction plot because you have, according to your description, held the variables with which they are interacted constant. A true interaction plot would have not just two survival curves but many pairs of such curves. Each pair would consist of one with high profitability and the other with low profitability, and each pair would correspond to a different value of, say, collective bargaining or individual rights (or perhaps some combinations of values of those). (Such a graph would probably be difficult to read, so I'm not sure it would be a good idea to do it anyway, but that's what it would be.)

Finally, you don't show any of the code. You don't really show the results either: you show how they look after they have been "pretty printed." So not being able to see those, I'll just point out that if you didn't use factor variable notation in your -stcox- command for the interactions, then your -stcurve- graphs are probably wrong anyway.

[Cheng Li]

There is actually nothing even remotely linear in this context.

Let's review the Cox proportional hazards model. First there is some "baseline" survival function (probability that the duration of a GFA, whatever that is, is at least amount of time t.) By "baseline" we mean the survival function that describes duration of a GFA when all of the covariates in the model are zero. The Cox proportional hazards model says that when you have non-zero covariates, then the survival function changes, by applying a constant ratio to the hazard function of the baseline survival curve. Recall that the hazard is -d(ln S)/dt where S is the survival function and t is time. Moreover the particular ratio to be applied to derive the hazard function is exp(b0 + b1X1 + b2X2 +... + bnXn) where the X's refer to the values of the model covariates, and the b's are the coefficients that -stcox- estimates for you. Notice that we have layer upon layer of nonlinearity here already. The only thing that is linear is the b0 + b1X1 + b2X2 + ... + bnXn, but between the exponentiation of that to derive the hazard ratio, and the logarithmic derivative relating the hazard to the survival function, that linearity is completely drowned out. And I haven't even mentioned that the baseline Survival function itself is usually highly non-linear to start with.

Your plots are graphs of S versus t with the various covariates held at the values you stipulated in your -stcurve- command, and shown separately for different values of profitability, and internationalization. As you can see from the previous paragraph it would be miraculous if any of these graphs were even close to linear. So miraculous that I have never seen that happen, and if I ever did, my first reaction would be that there must be some pretty serious errors in the data or the coding.

In addition, I note that although your full model contains interaction terms involving profitability and internationalization, neither of these graphs could be called an interaction plot because you have, according to your description, held the variables with which they are interacted constant. A true interaction plot would have not just two survival curves but many pairs of such curves. Each pair would consist of one with high profitability and the other with low profitability, and each pair would correspond to a different value of, say, collective bargaining or individual rights (or perhaps some combinations of values of those). (Such a graph would probably be difficult to read, so I'm not sure it would be a good idea to do it anyway, but that's what it would be.)

Finally, you don't show any of the code. You don't really show the results either: you show how they look after they have been "pretty printed." So not being able to see those, I'll just point out that if you didn't use factor variable notation in your -stcox- command for the interactions, then your -stcurve- graphs are probably wrong anyway.

Thank you for your feedback.
Here is my code for one set of the results (model 5) and the internationalization plot.

stcox code c.ud##c.taratio mining construction manufacturing transportation finance service age ln_gdppc gdpgr ln_noe ungc public sr hs sub
sum taratio
display taratio+r(sd)
display taratio-r(sd)
stcurve, survival at1(taratio=.89893983) at2(taratio=1.0847116) legend(label(1 "Low internationalization") label(2 "High internationalization"))

I forgot to mention I had plotted the graph by using two continuous variables.

In model 2, collective bargaining power variable has coefficient larger than 1, thus indicating an accelerating effect on GFAs signatory. In model 5, internationalization flipped the effects. However, I do not see the twist in the interaction plot. Am I missing something or asking the wrong question? Any help on how to interpret the results would be appreciated.

Thanks again,
Cheng

[Clyde Schechter]

In model 2, collective bargaining power variable has coefficient larger than 1, thus indicating an accelerating effect on GFAs signatory. In model 5, internationalization flipped the effects. However, I do not see the twist in the interaction plot. Am I missing something or asking the wrong question?

You are misinterpreting how the interaction terms in model 5 work.

There is no term in model 5 that corresponds to the collective bargaining power term in model 2. In model 2, your model assumes (imposes) the assumption that there is a single effect of collective bargaining power that applies regardless of internationalization. In model 5, by contrast, you assume there is no such thing at all. Rather model 5 assumes that the effect of collective bargaining power is different for different values of internationalization. The term in model 5 that is called collective bargaining power is actually the effect of collective bargaining power conditional on internationalization = 0. This may or may not be of any interest. In fact, if internationalization is defined in such a way that it cannot be 0, it is an estimate of the effect of collective bargaining power under conditions that are, in principle, impossible. Given that mean - 1 SD of internationalization is about 0.9 (and mean + 1 SD is about 1.1), the SD of internationalization must be something close to 0.1, and the mean must be close to 0, so internationalization = 0 would be something like 9 SD below the mean: so even if it is, in principle, possible, it must be very uncommon to see. So I suspect that the results in the collective bargaining power row for model 5 in your output table are probably not very useful.

The results shown for the interaction term of internationalization with collective bargaining power refer to the ratio of hazard ratios. That is, that 0.601 number means something like this: each unit increase in internationalization is associated with a reduction of the effect of collective bargaining power by a factor of 0.601. More generally, when internationalization takes on value x, the hazard ratio for collective bargaining power will be 1.681 * 0.601^x.

Nothing like this can be seen in the graphs you show because in these graphs, collective bargaining power is constrained at its mean value, and so its effects cannot be seen in the graph at all (let alone the difference in its effects at different values of internationalization.)

I am reluctant to go farther in trying to explain these graphs and results because the model 5 result for internationalization makes no sense. It is several orders of magnitude away from all of the other hazard ratios shown in that model. And I suspect it results from something odd in the data, perhaps a situation where there no, or almost no, non-censored outcomes in the observations where internationalization takes on low values, or some ranges of values of internationalization and collective bargaining power for which there are not many observations, but in those observations all the GFA durations are extremely short, or something like that. I don't think you should work with these results until that is resolved.

If I were in your situation, given that your sample size appears reasonably large, I would partition the ranges of both collective bargaining power and internationalization into, say, deciles and look at the mean and range of GFA durations in each of the 100 combination of deciles and see if there is something anomalous. By anomalous, I mean a cell or cells where there are either very few or no non-censored observations or where all of the GFA durations in the cell are very short.

[Cheng Li]

You are misinterpreting how the interaction terms in model 5 work.


There is no term in model 5 that corresponds to the collective bargaining power term in model 2. In model 2, your model assumes (imposes) the assumption that there is a single effect of collective bargaining power that applies regardless of internationalization. In model 5, by contrast, you assume there is no such thing at all. Rather model 5 assumes that the effect of collective bargaining power is different for different values of internationalization. The term in model 5 that is called collective bargaining power is actually the effect of collective bargaining power conditional on internationalization = 0. This may or may not be of any interest. In fact, if internationalization is defined in such a way that it cannot be 0, it is an estimate of the effect of collective bargaining power under conditions that are, in principle, impossible. Given that mean - 1 SD of internationalization is about 0.9 (and mean + 1 SD is about 1.1), the SD of internationalization must be something close to 0.1, and the mean must be close to 0, so internationalization = 0 would be something like 9 SD below the mean: so even if it is, in principle, possible, it must be very uncommon to see. So I suspect that the results in the collective bargaining power row for model 5 in your output table are probably not very useful.


The results shown for the interaction term of internationalization with collective bargaining power refer to the ratio of hazard ratios. That is, that 0.601 number means something like this: each unit increase in internationalization is associated with a reduction of the effect of collective bargaining power by a factor of 0.601. More generally, when internationalization takes on value x, the hazard ratio for collective bargaining power will be 1.681 * 0.601^x.


Nothing like this can be seen in the graphs you show because in these graphs, collective bargaining power is constrained at its mean value, and so its effects cannot be seen in the graph at all (let alone the difference in its effects at different values of internationalization.)


I am reluctant to go farther in trying to explain these graphs and results because the model 5 result for internationalization makes no sense. It is several orders of magnitude away from all of the other hazard ratios shown in that model. And I suspect it results from something odd in the data, perhaps a situation where there no, or almost no, non-censored outcomes in the observations where internationalization takes on low values, or some ranges of values of internationalization and collective bargaining power for which there are not many observations, but in those observations all the GFA durations are extremely short, or something like that. I don't think you should work with these results until that is resolved.


If I were in your situation, given that your sample size appears reasonably large, I would partition the ranges of both collective bargaining power and internationalization into, say, deciles and look at the mean and range of GFA durations in each of the 100 combination of deciles and see if there is something anomalous. By anomalous, I mean a cell or cells where there are either very few or no non-censored observations or where all of the GFA durations in the cell are very short.

Thank you, Dr. Clyde.

I went back and checked the underlying data for the internationalization variable and have found that internationalization was measured on a scale from 0 to 1 while my other ratio variables, i.e., solvency ratio and profitability, are from 1 to 100. After I converted internationalization by multiplying it by 100, the coefficient of internationalization as well as the interaction term —Collective bargaining legitimacy * Internalization — changed in model 5, as you can see below.

Can you please elaborate more on how to interpret the term in M5 and how to generate a sensible graph based on the results of M5.

Thanks, Cheng

[Clyde Schechter]

These results look a lot better. I think that rather than plotting survival curves at various values, I would graph the hazard ratios. The easiest way to do this is with -margins- and -marginsplot-. So following your -stcox- I would do something like this:

First since both ud and intl, as I understand it, now range between 0 and 100, I would do this

margins, at(ud = (10(10)90) intl = (10(10)90))
marginsplot, xdimension(intl) name(hr_intl_by_ud, replace)

margins, at(ud = (10(10)90) intl = (10(10)90))
marginsplot, xdimension(ud) name(hr_ud_by_intl, replace)

This will give you graphs of the hazard ratio as a function of intl at various levels of ud, and as a function of ud at various levels of int, and I think that will provide more insight into what is going on than any survival function plots would.

Remember that -stcox- starts with some baseline hazard function corresponding to both ud and intl = 0. Then relative to that, the hazard function associated with other values of ud and intl would be the baseline hazard function *1.229^intl * 1.759^ud * 0.994^ud*intl. The implication is that increasing values of intl or ud both lead to increasing hazard (i.e. steeper declines in the survival function for GFA duration, shorter GFA durations) but that the combination of the effects of intl and ud is slightly less than the separate effects would be. That is, there is a bit of interference between their effects: the whole is a bit less than the "sum" (really, in this case, product) of its parts.

Added: You may not need to run that same -margins- command twice. You may be able to do the two marginsplots one after the other. Be sure to use the -saving()- or -name()- option with your -marginsplot- commands so that the second graph doesn't overwrite the first. In earlier versions of Stata you had to re-run -margins- before you could do another -marginsplot-. In current Stata, however, you don't need to do that. Since running -margins- can take a long time if the data set is large, I recommend trying the code without the second -margins- command. If the second -marginsplot- fails with an error message that the previous command was not -margins-, then you need to rerun -margins- before doing the second graph.

More Added: Also bear in mind that a hazard ratio > 1 means an increasing hazard, which in turn means decreasing GFA durations. Similarly a hazard ratio < 1 means a decreasing hazard, which in turn means increasing GFA durations.

[Cheng Li]

These results look a lot better. I think that rather than plotting survival curves at various values, I would graph the hazard ratios. The easiest way to do this is with -margins- and -marginsplot-. So following your -stcox- I would do something like this:

First since both ud and intl, as I understand it, now range between 0 and 100, I would do this

margins, at(ud = (10(10)90) intl = (10(10)90))
marginsplot, xdimension(intl) name(hr_intl_by_ud, replace)

margins, at(ud = (10(10)90) intl = (10(10)90))
marginsplot, xdimension(ud) name(hr_ud_by_intl, replace)

This will give you graphs of the hazard ratio as a function of intl at various levels of ud, and as a function of ud at various levels of int, and I think that will provide more insight into what is going on than any survival function plots would.

Remember that -stcox- starts with some baseline hazard function corresponding to both ud and intl = 0. Then relative to that, the hazard function associated with other values of ud and intl would be the baseline hazard function *1.229^intl * 1.759^ud * 0.994^ud*intl. The implication is that increasing values of intl or ud both lead to increasing hazard (i.e. steeper declines in the survival function for GFA duration, shorter GFA durations) but that the combination of the effects of intl and ud is slightly less than the separate effects would be. That is, there is a bit of interference between their effects: the whole is a bit less than the "sum" (really, in this case, product) of its parts.

Added: You may not need to run that same -margins- command twice. You may be able to do the two marginsplots one after the other. Be sure to use the -saving()- or -name()- option with your -marginsplot- commands so that the second graph doesn't overwrite the first. In earlier versions of Stata you had to re-run -margins- before you could do another -marginsplot-. In current Stata, however, you don't need to do that. Since running -margins- can take a long time if the data set is large, I recommend trying the code without the second -margins- command. If the second -marginsplot- fails with an error message that the previous command was not -margins-, then you need to rerun -margins- before doing the second graph.

More Added: Also bear in mind that a hazard ratio > 1 means an increasing hazard, which in turn means decreasing GFA durations. Similarly a hazard ratio < 1 means a decreasing hazard, which in turn means increasing GFA durations.

Thank you, Dr. Clyde. I intend to interpret the model 5 results as below. Any suggestion?

The coefficient of interaction term between Collective Bargaining Legitimacy and Internationalization is 0.995 (p =0.005), which is below 1 and hence indicates later signing. Thus, Model 5 suggests that the propensity of Collective Bargaining Legitimacy on MNEs commitment to a GFA is 4.6% smaller for firms with one more standard deviation of international assets (all other variables held constant).

note: one standard deviation of Internationalization equals to 9.288587. I calculate 4.6% = 9.288587*(1-0.995).

Thanks again, Cheng

[Clyde Schechter]

Unfortunately, that interpretation is incorrect in several material ways.

First, the 0.995 number is not the coefficient of the interaction term, it is the ratio of hazard ratios associated with the interaction. Because it is associated with an interaction term, rather than a "main" effect, it does not represent either earlier or later signing. Rather, being less than 1, it indicates that the combination of a given degree of collective bargaining legitimacy and international assets does not shorten the time to signing by as much as would be seen if collective bargaining legitimacty and international assets affected time to signing independently of each other. Instead, the two effects interfere somewhat with each other.

The 4.6% figure is computed by a wrong formula, although it turns out to be not far from the correct value because (1-x)^y is approximately equal to 1 - y*x when x is, as here, very small. The contribution of the interaction to the overall hazard ratio associated with one standard deviation of internationalization is 0.995^9.288587 = 0.955 (to 3 decimal places). So with a 1sd difference in international assets, the impact (in the hazard ratio metric) of interntionalization is attenuated by a factor of 1-0.955 = 4.5% relative to what it would be if collective bargaining legitimacy and internationalization affected time to signing independently. This number cannot be interpreted as an effect on time to signing itself. It is the amount by which the effect on time to signing is modified due to interaction.

[Cheng Li]

Unfortunately, that interpretation is incorrect in several material ways.

First, the 0.995 number is not the coefficient of the interaction term, it is the ratio of hazard ratios associated with the interaction. Because it is associated with an interaction term, rather than a "main" effect, it does not represent either earlier or later signing. Rather, being less than 1, it indicates that the combination of a given degree of collective bargaining legitimacy and international assets does not shorten the time to signing by as much as would be seen if collective bargaining legitimacty and international assets affected time to signing independently of each other. Instead, the two effects interfere somewhat with each other.

The 4.6% figure is computed by a wrong formula, although it turns out to be not far from the correct value because (1-x)^y is approximately equal to 1 - y*x when x is, as here, very small. The contribution of the interaction to the overall hazard ratio associated with one standard deviation of internationalization is 0.995^9.288587 = 0.955 (to 3 decimal places). So with a 1sd difference in international assets, the impact (in the hazard ratio metric) of interntionalization is attenuated by a factor of 1-0.955 = 4.5% relative to what it would be if collective bargaining legitimacy and internationalization affected time to signing independently. This number cannot be interpreted as an effect on time to signing itself. It is the amount by which the effect on time to signing is modified due to interaction.

Thank you, Dr. Clyde.
You have been very helpful. I am a first year business school Ph.D student and eager to learn some good methods which I can apply to different research settings. Any good stats or method books
you would recommend me to read?

Again, thanks a lot,
Cheng

[Cheng Li]

Unfortunately, that interpretation is incorrect in several material ways.

First, the 0.995 number is not the coefficient of the interaction term, it is the ratio of hazard ratios associated with the interaction. Because it is associated with an interaction term, rather than a "main" effect, it does not represent either earlier or later signing. Rather, being less than 1, it indicates that the combination of a given degree of collective bargaining legitimacy and international assets does not shorten the time to signing by as much as would be seen if collective bargaining legitimacty and international assets affected time to signing independently of each other. Instead, the two effects interfere somewhat with each other.

The 4.6% figure is computed by a wrong formula, although it turns out to be not far from the correct value because (1-x)^y is approximately equal to 1 - y*x when x is, as here, very small. The contribution of the interaction to the overall hazard ratio associated with one standard deviation of internationalization is 0.995^9.288587 = 0.955 (to 3 decimal places). So with a 1sd difference in international assets, the impact (in the hazard ratio metric) of interntionalization is attenuated by a factor of 1-0.955 = 4.5% relative to what it would be if collective bargaining legitimacy and internationalization affected time to signing independently. This number cannot be interpreted as an effect on time to signing itself. It is the amount by which the effect on time to signing is modified due to interaction.

.


Thank you, Dr. Clyde. After reading your posts. I intend to interpret my results as below. Any suggestion?

Model 2 indicates that a one standard deviation increase in Collective Bargaining Legitimacy (all other variables held constant) enhances the rate of substantive action by a factor of 2.568 (=1.061^15.93 = beta( collective bargaining legitimacy)^SD(collective bargaining legitimacy) to 3 decimal places ).

In Model 3, the Hazard ratio of Individual Employee Rights is 2.477 and significant at p=0.000. Thus, a one standard deviation increases in individual rights of employees (all other variables held constant) accelerates the rate of commitment by a factor of 2.513(=2.477^1.016 = beta (individual Employee Rights)^SD(individual Employee Rights) to 3 decimal places ).

Results in Model 4 indicate the rate of commitment increases by a rate of 28.46 for firms in construction sectors than in Wholesale and retail trade, the baseline industry for the industry dummies (all other variables held constant).

Model 5 suggests that the contribution of the interaction to the overall hazard ratio associated with one standard deviation of internationalization is 0.955 (=0.9959.288587 = beta(collective bargaining legitimacy*internationalization)SD(internationaliz ation); to 3 decimal places).

In Model 6, we find that the magnitude of hazard ratio of the interaction term between Individual Employment Rights and Internationalization is big (13.54) but not significant. Hence, we cannot retain H4b.

In Model 7, the coefficient of interaction term between Collective Bargaining Legitimacy and Profitability is not significant.

In Model 8, we find the hazard ratio of interaction term between Individual Employment Rights and Profitability is 1.093 and significant (p = 0.019). This result suggests that the contribution of the interaction to the overall hazard ratio associated with one standard deviation of profitability is 1.925 (=1.0937.362=beta (collective bargaining legitimacy*profitability)^SD(profitability); to 3 decimal places).

Also while I was reading papers on one of the top management journals, I have found that some papers interpreted coefficient of interaction term as an effect on time/duration to dependent variable of Cox model. One example can be seen below.

“ … The interaction between Technology Convergence and the incumbent dummy yields a highly significant positive effect: the coefficient for Incumbent * Technology Convergence in model 5 is b= 9.534,p< .001. In terms of effect size, a one-standard-deviation increase in technology closeness, when Incumbent is 1, exp ((– 10.86 1 9.534) * 0.128) = 0.84 (84% of the base hazard rate), decreases industry incumbents’ risk of repositioning through fab closure by 16%.

… Consistent with Hypothesis 3, the coefficient for the interaction term is positive and highly significant for entrants (b = 78.53, p <001) and negative and highly significant for industry incumbents (b =-26.94, p ,<.001). In terms of effect size, a one standard- deviation increase in market closeness, when technology closeness is at its maximum (0.86), exp((–66.65 +0.86 * 78.53) * 0.07) =1.06 (106% of the base hazard rate), increases entrants’ risk of exit by 6%.

… For industry incumbents, the same increase in market closeness, when technology closeness is at its maximum, exp((20.28 + 0.86 * -26.94) * 0.07) = 0.82 (82% of the base hazard rate), decreases industry incumbents’ risk of fab closure by 18%. ”

I have attached the relevant part (M5) below. Please help to shed some light on this matter as I am very confused.

[Clyde Schechter]

So my first question is why you are doing these things in terms of a 1 SD change in the predictor variable. Do these predictor variables have no intelligble scale or units of their own? If so, I suppose a 1 SD unit conveys something that people might grasp. But if these variables (and I am not familiar with this field and know nothing about these variables) have units or a scale of their own, going to standard deviations just obfuscates the results. So if they have units of their own, it is much clearer to just report results in terms of the original units.

In referring to the results for an interaction term, they are not hazard ratios, they are ratios of hazard ratios.

You should not describe or interpret Cox proportional hazards results in terms of "risk." The term "risk of exit" (or fab closure, or whatever) refers to the probability that an exit will eventually occur. The relationship between a hazard ratio and this probability is extremely complicated and calculating it from the results of a Cox regression would be very difficult, and often not even possible. I think the reason you can find examples of this in the literature is that hazard ratios are complicated and many people use them without really understanding them. But claims about "risk" from a Cox analysis are almost always incorrect, unless some really elaborate calculations have been undertaken. What a hazard ratio analysis tells you is the change in the instantaneous rate at which something happens, not the risk that it will happen. I suppose one might charitably call the use of "risk" in this context an abuse of language, but it is at least as incorrect as speaking of a distance traveled when you are really talking about a velocity.