{smcl}
{* 30nov2005}{...}
{hline}
help for {hi:stkerhaz}
{hline}

{title:Baseline Hazard Estimates via Kernel Smoother and plots}

{p 4 13}{cmd:stkerhaz} [{cmd:if} {it:exp}] [{cmd:in} {it:range}] {cmd:,} {cmdab:b:width(}{it:# # #}{cmd:)}
[ {cmdab:k:erkode(}{it:#}{cmd:)} {cmdab:np:oint(}{it:#}{cmd:)} {cmdab:basech:a(}{it:varname}{cmd:)} 
{cmd:strata(}{it:varname}{cmd:)} {cmd:tmax} {cmd:ci} {cmdab:l:evel(}{it:#}{cmd:)}
{cmdab:out:file(}{it:filename[,replace]}{cmd:)} {cmd:per(}{it:#}{cmd:)} {it:graph_options} ]


{p}{cmd:stkerhaz} is for use with survival-time data; see help {help st}. You must 
have {cmd:stset} your data before using this command; see help {help stset}.{p_end} 
{p}{cmd:stkerhaz} needs {help levels7} which you can download by 
{net "describe http://www.stata.com/stb/stb60/dm90"}. 
See {help findit} and {help ssc install} to install user-written additions from the net.


Examples:

{p 4 4}Baseline Hazard Plot with 95% confidence bounds{p_end}
{p 12 20}{inp:. stcox, estimate basech(H)}{p_end}
{p 12 20}{inp:. stkerhaz,bwidth(10) ci}

{p 4 4}Baseline Hazard Plot with two curves for 5 and 10 time units bandwidth{p_end}
{p 12 20}{inp:. stcox, estimate basech(H)}{p_end}
{p 12 20}{inp:. stkerhaz,bwidth(5 10)}

{p 4 4}Baseline Hazard Plot with as much curves as hormon values. The estimates are saved in
\currentfolder\myhaz{p_end}
{p 12 20}{inp:. stcox, estimate basech(H) strata(hormon)}{p_end}
{p 4}or {break}{p_end}
{p 12 20}{inp:. sts gen H = na, by(hormon)}{p_end}
{p 12 20}{inp:. stkerhaz,bwidth(10) strata(hormon) out(myhaz)}

{p 4 4}Adjusted Baseline SMR Plot{p_end}
{p 12 20}{inp:. xi: stcox i.yfecat i.afecat, basech(H) offset(lograte)}{p_end}
{p 12 20}{inp:. stkerhaz,b(5)}


{title:Description}

{p}{cmd:stkerhaz} computes nonparametric estimates of the baseline hazard or baseline SMR and draws 
the graph of the results. This command can be used after {cmd:stcox}. In this case it requires that 
you previously specified {cmd:stcox}'s {cmd:basech()} option; see {help stcox}.
Otherwise in {cmd:stkerhaz}'s {cmdab:basech:a} option you can specify {it:varname} storing cumulative 
baseline hazard. {break}
Actually {cmd:stkerhaz} can be used even to smooth a cumulative excess mortality function, so achieving
a smoothed estimate of excess mortality function.


{title:Options}

{p 0 4}{cmd:bwidth(}{it:# # # #}{cmd:)} specifies the window half-width to be used. {cmd:bwidth} is not
optional. At least one bandwidth must be specified. Up to four bandwidths can be used.
Then, curves are drawn for each bandwidth.

{p 0 12}{cmd:kerkode(}{it:#}{cmd:)} specifies the weight function (kernel) to be used according to the
following numerical codes: 
	{break}	1 = Uniform 		   
	{break} 2 = Epanechnikov (Default) 
	{break} 3 = Biweight		   

Asymmetric kernel (see {title:Remarks} below) is computed where appropriate.

{p 0 4}{cmd:npoint(}{it:#}{cmd:)} specifies the number of equally spaced points in the range used for
the estimation. Default is 150. Unless {cmd:tmax} is used, the range of points starts at the lowest {it:_t0} 
and stops at the last death time. 

{p 0 4}{cmd:basecha(}{it:varname}{cmd:)} specifies the variable storing baseline cumulative hazard.

{p 0 4}{cmd:strata(}{it:varname}{cmd:)} option is intended for use in conjunction with the {cmd:strata} 
option of {cmd:stcox} or {cmd:by} option of {cmd:sts gen}. It enables to calculate and graph up to 
four baseline hazard curves for corresponding numeric values of strata or by variable. If there are more
values of the strata variable it needs to save estimates in a file and then graph them as wished. 

{p 0 4}{cmd:tmax} sets the starting point of range at the time of the earliest death.

{p 0 4}{cmd:ci} allows to plot confidence bounds around baseline smoothed hazard. Note that they are
correct only when Baseline Cumulative Hazard derives from an unadjusted model. Multiple curves and 
{cmd:ci} cannot be plotted at once.

{p 0 4}{cmd:level(}{it:#}{cmd:)} specifies the confidence level, in percent, for the pointwise confidence bounds. Default is 95.

{p 0 4}{cmd:outfile(}{it:filename[,replace]}{cmd:)} saves in filename values used to plot. The variable 
containing baseline hazard estimates is prefixed by KS_. If confidence bounds are plotted the standard 
error and pointwise high and low confidence interval, based on a log transformation, are saved as well 
in the variables prefixed by KS_SE_, KS_HI_ and KS_LO_. Rest of names specifies bandwidth or value 
of stratavar which they refer to. A variable named Gridpoint saves equally spaced data points where
estimates are calculated.

{p 0 4}{cmd:per(}{it:#}{cmd:)} defines the time units in which the estimated hazards will be reported. 
If the time analysis is in year, specifying per(1000) results in the graph are in rates per 1000 
person-years.


{title:Remarks}

{p}The kernel-smoothed hazard estimated by {cmd:stkerhaz} uses the method described by  Breslow and
Day (1986, pp 178-229) and by Klein and Moeschberger (2003, pp 166-177) first introduced by Ramlau-Hansen
(see formula 5.18). {break}
This estimate is simply a weighted average of the increments in cumulative baseline hazard,
where weights are a kernel function of {it: ((t_target - t_obs) / bandwidth)} defined in the interval
{it: [-1,+1]}. {it: t_target} are time points at which baseline hazard is estimated and {it: t_obs} 
are times at which increment of cumulative baseline hazard is actually observed. Kernel weigths are defined as:{p_end}
	Uniform      {col 25} K(z) = 0.5 if |z| < 1 and K(z) = 0 otherwise            
	Epanechnicov {col 25} K(z) = 0.75(1-z^2) if |z| < 1 and K(z) = 0 otherwise    
	Biweight     {col 25} K(z) = 0.975(1-z^2)^2 if |z| < 1 and K(z) = 0 otherwise 

{p}When {it: t_target} or, in the right-hand tail, {it: (last_death_time - t_target)} is smaller 
than {it:bandwidth} asymmetric kernels are computed according to the formulas in Klein and 
Moeschberger's book (2003, pp 167-168).

{p}As pointed out by Breslow and Day, an estimate of cumulative baseline SMR can be obtained incorporating
into a Cox model as an offset term the logarithm of time-dependent standard rates. 
Cumulative SMR can be smoothed using the same method to yield non parametric estimates of SMR at various
points in time analysis axis.

{p}They say: "Cumulative baseline mortality or incidence rates are not as informative as they might appear
at first sight. They tend to overemphasize the jumps at very high times at which the
estimate is least stable. Also, time-specific rates are usually of greater intrinsic interest than the
cumulative rate." Furthermore, graphing baseline hazard or SMR, trend along time axis can be appreciated showing when a
rise or a decline appears and if it is sharp or gradual.
{cmd:stcox} or {cmd:sts gen} can create new variable containing cumulative baseline hazard unadjusted or adjusted
for covariates or estimated in separate groups. This command is aimed at easily deriving from this stored
result an estimate of the baseline hazard.

{p}Klein and Moeschberger also advice to use the kernel smoothing technique to compute a smoothed estimate
of the excess mortality function starting from a cumulative excess mortality function.

{p}Bandwidths have to be chosen being aware that small bandwidth (small smoothing) yields hazard (or SMR) estimates
affected by random noise, while large bandwidth (large smoothing) blurs the structure of the data.
Visual appearance of the graph can address the selection, although objective criteria exist.



{title:Also see}

{p 1 10}Manual:  {hi:{bind:[R] kdensity}}, {hi:{bind:[R] st stcox}}
{hi:{bind:[R] st sts generate}}, {hi:{bind:[R] st sts graph}}{p_end}
{p 0 19}On-line:  help for {help kernreg}, {help bhcalc}, {help sthaz} if installed{p_end}


{title:Note}

{p} A previous version of stkerhaz computed more kinds of kernel weigths, but it did not compute
asymmetric kernel in the tails of the analysis time. On request I can provide this old version. 


{title:References}

{p} Breslow, N. E. and Day, N. E.  Statistical Methods in Cancer Research. Volume II - The Design and
analysis of cohort studies.  Lyon: International Agency for Research on Cancer, 1987.

{p} Klein, J. P. and Moeschberger, M. L. Survival Analysis Techniques fo Censored and Truncated Data (2nd
Edition). New York: Springer-Verlag, 2003.


{title:Author}

        Enzo Coviello, Azienda U.S.L. BA/1, Italy
        enzo.coviello@tin.it