JuliaActuary
Practical, extensible, and open-source actuarial modeling and analysis.

Fitting survival data with MortaltityTables.jl

via PharmCat on Github (this link has similar code with comments in Russian and English).

begin 
    using LsqFit
    using MortalityTables
    using Plots
    using Distributions
    using Optim
    using DataFrames
    using PlutoUI; TableOfContents()
end

Fitting a Weibull survival curve

Sample data:

data = let
    survival = [0.99,0.98,0.95,0.9,0.8,0.65,0.5,0.38,0.25,0.2,0.1,0.05,0.02,0.01]
    times = 1:length(survival)
    DataFrame(;times,survival)
end
times survival
1 1 0.99
2 2 0.98
3 3 0.95
4 4 0.9
5 5 0.8
6 6 0.65
7 7 0.5
8 8 0.38
9 9 0.25
10 10 0.2
11 11 0.1
12 12 0.05
13 13 0.02
14 14 0.01 
plt = plot(data.times,data.survival,label="observed survival proportion",xlabel="time")

Define the two-parameter Weibull model:

model(x, p) will accept the full data set as the first argument x. This means that we need to write our model function so it applies the model to the full dataset. We use @. to apply ("broadcast") the calculations across all rows.

@. model1(x, p) = survival(MortalityTables.Weibull(;m = p[1],σ = p[2]), x)
model1 (generic function with 1 method)

Fitting the Model

And fit the model with LsqFit.jl:

fit1 = curve_fit(model1, data.times, data.survival, [1.0, 1.0])
LsqFit.LsqFitResult{Vector{Float64}, Vector{Float64}, Matrix{Float64}, Vector{Float64}}([8.147278225549185, 2.8507079975810274], [0.007512289530956395, 0.0021036645286324207, -0.0059128358266353764, -0.02271535116992418, -0.019432525084171814, 0.008926964702135387, 0.02305783722832544, 0.007050778443488892, 0.01472116019643549, -0.03405121675050801, -0.005432126302967605, -0.0014077035272865995, 0.0023337166510039725, -0.0008907842989644022], [0.0026998722509579453 -0.005225236784169305; 0.014959447652454607 -0.024973364570363275; … ; 0.015866902563881607 0.03977450356606966; 0.00688563201685319 0.02322957839976213], true, Float64[])
plot!(plt,data.times, model1(data.times, fit1.param),label="fitted model")

Maximum Likelihood estimation

Generate 100 sample datapoints:

t = rand(Weibull(fit1.param[2], fit1.param[1]), 100)
100-element Vector{Float64}:
  9.702067663069327
  3.0428689574368324
  9.059639240991595
 10.970739445634079
  3.9040404619667113
  4.165794231001152
  4.842637489172447
  ⋮
  5.6539647956004515
  6.139331596615337
  9.641578582932453
  4.896864716185817
  4.434432566682164
 11.444165198474122

Without Censored Data

#No censored data
fit_mle(Weibull, t)
Distributions.Weibull{Float64}(α=2.852156865689702, θ=8.150334609678797)

With Censored Data

c = collect(trues(100))
100-element Vector{Bool}:
 1
 1
 1
 1
 1
 1
 1
 ⋮
 1
 1
 1
 1
 1
 1
c[[1,3,7,9]] .= false
4-element view(::Vector{Bool}, [1, 3, 7, 9]) with eltype Bool:
 0
 0
 0
 0
#ML function
survmle(x) = begin
    ml = 0.0
    for i = 1:length(t)
        if c[i]
            ml += logpdf(Weibull(x[2], x[1]), t[i]) #if not censored log(f(x))
        else
            ml += logccdf(Weibull(x[2], x[1]), t[i]) #if censored log(1-F)
        end
    end
    -ml
end
survmle (generic function with 1 method)

opt = Optim.optimize(
    survmle,          	# function to optimize
    [1.0,1.0], 			# lower bound
    [15.,15.],            # upper bound
    [3.,3.]          	# initial guess
)
 * Status: success

 * Candidate solution
    Final objective value:     2.379075e+02

 * Found with
    Algorithm:     Fminbox with L-BFGS

 * Convergence measures
    |x - x'|               = 9.46e-08 ≰ 0.0e+00
    |x - x'|/|x'|          = 1.08e-08 ≰ 0.0e+00
    |f(x) - f(x')|         = 0.00e+00 ≤ 0.0e+00
    |f(x) - f(x')|/|f(x')| = 0.00e+00 ≤ 0.0e+00
    |g(x)|                 = 5.75e-09 ≤ 1.0e-08

 * Work counters
    Seconds run:   1  (vs limit Inf)
    Iterations:    4
    f(x) calls:    50
    ∇f(x) calls:   50

The solution converges to similar values as the function generating the synthetic data:

Optim.minimizer(opt)
2-element Vector{Float64}:
 8.26940849837394
 2.8564783058448087

Fitting Kaplan Meier

KaplanMeier comes from Survival.jl

#4
#Подгонка модели по эмпирической функции KM
#Fitting for survival function from Kaplan Meier

using Survival
#t- time vector;c - censored events vector
km = fit(Survival.KaplanMeier, t, c)
KaplanMeier{Float64}([1.5497716226250795, 1.6775379881561192, 1.7085009082750642, 2.5181132678997726, 2.68032348518394, 2.964498811102057, 3.0428689574368324, 3.1715173900300746, 3.3408214242675918, 3.558112309435965  …  10.933124905309146, 10.968571881614759, 10.970739445634079, 11.237243842252303, 11.343262031763562, 11.444165198474122, 11.53918544474052, 12.139001873035417, 12.760811345926422, 13.990010616742882], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1  …  1, 1, 1, 1, 1, 1, 1, 1, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0  …  0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [100, 99, 98, 97, 96, 95, 94, 93, 92, 91  …  10, 9, 8, 7, 6, 5, 4, 3, 2, 1], [0.99, 0.98, 0.97, 0.96, 0.95, 0.94, 0.9299999999999999, 0.9199999999999999, 0.9099999999999999, 0.8999999999999999  …  0.1000085820703387, 0.08889651739585662, 0.07778445272137455, 0.06667238804689248, 0.055560323372410396, 0.04444825869792832, 0.03333619402344624, 0.022224129348964162, 0.011112064674482081, 0.011112064674482081], [0.01005037815259212, 0.014285714285714285, 0.017586311452816476, 0.020412414523193152, 0.022941573387056175, 0.025264557631995567, 0.02743516305843672, 0.029488391230979426, 0.03144854510165755, 0.03333333333333333  …  0.3122238849169251, 0.33372540089346836, 0.35948544624038536, 0.39120238990600864, 0.4317089797554669, 0.4861816977237071, 0.5654254827427213, 0.6974042179407192, 0.9931629489673415, 0.9931629489673415])
plt2 = plot(km.times, km.survival; labels="Empirical")
@. model(x, p) = survival(MortalityTables.Weibull(;m = p[1],σ = p[2]), x)
model (generic function with 1 method)
mfit = LsqFit.curve_fit(model, km.times, km.survival, [2.0, 2.0])
LsqFit.LsqFitResult{Vector{Float64}, Vector{Float64}, Matrix{Float64}, Vector{Float64}}([8.378620456975263, 3.2256169578710185], [-0.002403778609877949, 0.0047839066859725055, 0.014049535027020887, -0.0030835485933451423, -0.0004737254028512883, -0.005074670900177991, 0.0005240823447953957, 0.0029460936677042637, 0.002303863238325876, -0.0024660638160870496  …  0.03584410203827022, 0.044685608616582315, 0.05565969008761626, 0.05055522784081323, 0.05562383494771387, 0.06119548794504302, 0.0672761069565029, 0.050610626595534725, 0.039553131229845835, 0.011545102549992596], [0.010270434299658012 -0.01675139926361279; 0.01221021915932723 -0.019556832019687578; … ; 0.027138545352528534 0.05119262536384386; 0.012964385596268838 0.035424971300125514], true, Float64[])
plot!(plt2,km.times, model(km.times, mfit.param), labels="Theoretical")

Built with Julia 1.8.5 and

DataFrames 1.3.2
Distributions 0.25.49
LsqFit 0.12.1
MortalityTables 2.2.0
Optim 1.6.1
Plots 1.25.11
PlutoUI 0.7.34
Survival 0.2.2

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