Calculate hazard from digitized survival curves

Question

I have digitized published survival curves and got the following (Survival,time) points

df<-structure(list(Survival = c(1, 1, 1, 1, 1, 1, 0.99, 0.99, 0.99, 
0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 
0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.98, 0.98, 0.98, 0.98, 
0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 
0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 0.98, 
0.98, 0.98, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 
0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 
0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 
0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 0.97, 
0.97, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 
0.96, 0.96, 0.96, 0.96, 0.96, 0.96, 0.95, 0.95, 0.95, 0.95, 0.95, 
0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 
0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 
0.95, 0.95, 0.95, 0.95, 0.95, 0.95, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 0.94, 
0.94, 0.94, 0.94, 0.94, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 0.93, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 
0.92, 0.92, 0.92, 0.92, 0.92, 0.92, 0.91, 0.91, 0.91, 0.91, 0.91, 
0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 
0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 
0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 
0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.91, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 0.9, 
0.9, 0.9, 0.9, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 
0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 0.89, 
0.89, 0.89, 0.89, 0.89, 0.89), time = c(0, 0.34, 0.55, 0.82, 
0.87, 0.87, 0.87, 0.87, 0.87, 1.02, 1.02, 1.24, 1.45, 1.61, 1.74, 
1.83, 1.86, 1.87, 1.87, 1.87, 1.88, 1.96, 2.16, 2.37, 2.75, 3.01, 
3.48, 4.27, 4.5, 4.68, 4.89, 5.19, 5.58, 6.43, 6.75, 7.11, 7.55, 
7.99, 8.41, 8.82, 9.26, 9.72, 10.09, 10.41, 10.68, 10.99, 11.29, 
11.58, 11.85, 12.2, 12.47, 12.69, 13.03, 13.31, 13.72, 14.18, 
14.48, 14.79, 15.26, 15.83, 16.08, 16.55, 16.92, 17.74, 18.54, 
19.24, 19.54, 19.92, 21.14, 21.51, 21.89, 22.98, 23.29, 23.55, 
23.78, 24.1, 24.38, 24.79, 25.09, 25.42, 25.78, 27.09, 27.66, 
28.18, 28.57, 28.97, 29.36, 29.76, 30.15, 30.54, 30.94, 31.33, 
31.73, 32.12, 32.64, 33.19, 33.56, 33.8, 34.05, 34.22, 34.5, 
34.79, 35.1, 35.61, 36.05, 36.47, 36.87, 37.26, 37.65, 38.05, 
38.44, 38.84, 39.23, 39.63, 40.02, 40.42, 40.81, 41.21, 41.6, 
41.99, 42.51, 43.1, 43.41, 43.98, 44.91, 45.55, 46.53, 47.44, 
47.83, 48.17, 49.03, 49.65, 50.45, 50.84, 51.22, 51.61, 52.4, 
52.85, 53.24, 53.62, 54.62, 55.11, 55.46, 56, 56.61, 57.39, 57.75, 
58.15, 58.47, 58.77, 59.01, 59.57, 59.94, 60.44, 60.95, 61.35, 
61.74, 62.14, 62.53, 62.93, 63.32, 63.71, 64.11, 64.62, 65.3, 
65.61, 65.96, 66.21, 66.71, 67.08, 67.6, 68.04, 68.45, 68.85, 
69.24, 69.64, 70.03, 70.43, 70.82, 71.22, 71.61, 72.01, 72.4, 
72.79, 73.19, 73.58, 73.98, 74.37, 74.79, 75.23, 75.68, 76.13, 
76.41, 76.65, 76.88, 77.12, 77.52, 77.88, 78.3, 78.8, 79.22, 
79.64, 79.95, 80.94, 81.32, 81.66, 82.5, 82.86, 83.24, 83.58, 
84.01, 84.27, 84.63, 84.94, 85.28, 86.25, 86.59, 86.91, 88.39, 
88.73, 89.3, 89.59, 90.12, 90.39, 90.67, 91.18, 91.7, 92.14, 
92.55, 92.95, 93.34, 93.74, 94.13, 94.53, 94.92, 95.31, 95.71, 
96.1, 96.5, 96.89, 97.29, 97.68, 98.08, 98.47, 98.87, 99.26, 
99.66, 100.05, 100.44, 100.84, 101.23, 101.63, 102.02, 102.42, 
102.81, 103.21, 103.6, 104, 104.39, 104.78, 105.18, 105.57, 105.97, 
106.36, 106.76, 107.15, 107.55, 107.94, 108.34, 108.73, 109.13, 
109.52, 109.91, 110.31, 110.7, 111.12, 111.56, 112.27, 112.64, 
113.1, 113.59, 114.1, 114.46, 114.8, 115.24, 115.55, 116, 116.34, 
117.03, 117.85, 118.24, 118.58, 118.89, 120.23, 120.55, 120.85, 
121.92, 122.4, 122.89, 123.33, 123.75, 124.14, 124.54, 124.93, 
125.33, 125.72, 126.11, 126.51, 126.9, 127.3, 127.69, 128.09, 
128.48, 128.88, 129.27, 129.67, 130.06, 130.45, 130.85, 131.24, 
131.64, 132.03, 132.43, 132.82, 133.22, 133.61, 134.01, 134.4, 
134.8, 135.19, 135.58, 135.98, 136.37, 136.77, 137.16, 137.56, 
137.95, 138.35, 138.74, 139.14, 139.53, 139.92, 140.34, 140.78, 
141.49, 141.86, 142.29, 142.73, 143.3, 143.81, 144.27, 144.73, 
145.23, 145.66, 146.01, 146.33, 146.92, 147.74, 148.26, 148.66, 
149.21, 149.93, 150.33, 150.87, 151.38, 151.78, 152.17, 152.57, 
152.96, 153.36, 153.75, 154.14, 154.54, 154.93, 155.33, 155.72, 
156.12, 156.51, 156.91, 157.3, 157.81, 158.49, 158.82, 159.33, 
159.8, 160.19, 160.74, 161.25, 161.76, 162.44, 162.76, 163.1, 
163.35, 163.98, 164.38, 164.9, 165.25, 165.57, 166.11, 167.09, 
167.48, 167.9, 168.34, 168.75, 169.15, 169.54, 169.94, 170.35, 
170.79, 171.51, 171.88, 172.31, 172.74, 173.32, 173.82, 174.26, 
174.68, 175.07, 175.47, 175.86, 176.26, 176.65, 177.04, 177.44, 
177.83, 178.23, 178.62, 179.02, 179.41, 179.81, 180.2, 180.6, 
180.99, 181.38, 181.78, 182.17, 182.57, 182.96, 183.36, 183.75, 
184.15, 184.54, 184.94, 185.33, 185.73, 186.12, 186.51, 186.91, 
187.3, 187.7, 188.09, 188.49, 188.88, 189.28, 189.67, 190.07, 
190.46, 190.85, 191.25, 191.64, 192.04, 192.43, 192.83, 193.22, 
193.62, 194.01, 194.41, 194.8, 195.2, 195.59, 195.98, 196.38, 
196.77, 197.17, 197.56, 197.96, 198.37, 198.81, 199.62, 199.91, 
200.33, 200.85, 201.34, 201.84, 202.28, 202.7, 203.09, 203.48, 
203.88, 204.27, 204.67, 205.06, 205.46, 205.85, 206.25, 206.64, 
207.04, 207.43, 207.83, 208.22, 208.61, 209.01, 209.4, 209.8, 
210.19, 210.59, 210.98, 211.38, 211.77, 212.17, 212.56, 212.95, 
213.37, 213.81, 214.33, 214.73, 215.63, 216.28, 216.79, 217.3, 
217.69, 218.09, 218.48, 218.88, 219.27, 219.67, 220.06, 220.46, 
220.85, 221.24, 221.64, 222.03, 222.43, 222.82, 223.22, 223.61, 
224.01, 224.4, 224.8, 225.19, 225.59, 225.98, 226.37, 226.77, 
227.16, 227.56, 227.95, 228.35, 228.74, 229.14, 229.53, 229.93, 
230.32, 230.71, 231.11, 231.5, 231.9, 232.29, 232.81, 233.34, 
233.77, 234.34, 235.23, 235.73, 236.24, 236.64, 237.03, 237.43, 
237.82, 238.22, 238.61, 239.4, 239.77, 240.07, 241.47, 242.05, 
242.56, 242.95, 243.35, 243.74, 244.14, 244.53, 244.93, 245.32, 
245.72, 246.11, 246.51, 246.9, 247.29, 247.69, 248.08, 248.48, 
248.87, 249.27, 249.66, 250.06, 250.45, 250.85, 251.24, 251.63, 
252.03, 252.42, 253.4, 253.99, 254.7, 255.15, 255.56, 256.09, 
256.54, 256.87, 257.19, 258.14, 258.48, 258.8, 259.18, 260.18, 
260.56, 261.05, 261.49, 261.9, 262.3, 262.69, 263.09, 263.48, 
264.14, 264.98, 265.49, 265.87, 266.22, 267.66, 268.15, 268.6, 
269.01, 269.41, 269.8, 270.19, 270.59, 270.98, 271.4, 271.84, 
272.36, 272.76, 273.66, 274.31, 274.82, 275.33, 275.72, 276.12, 
276.63, 277.32, 277.67, 277.91, 278.28, 278.54, 278.84, 279.16, 
279.56, 280, 280.73, 281.48, 282.04, 282.61, 283.31, 284.14, 
284.32, 284.44, 284.64, 284.88, 285.15, 285.45, 285.79, 286.17, 
286.72, 287.15, 287.48, 287.82, 288.35, 289.08, 290.04, 290.32, 
290.82, 291.32, 291.7, 292.2, 292.71, 293.42, 293.94, 294.56, 
295.29, 295.61, 296.27, 297.13, 297.45, 297.79, 298.22, 298.54, 
298.88, 299.41, 299.82, 300.22, 300.61, 301.01, 301.4, 301.8, 
302.19, 302.58, 302.98, 303.37, 303.77, 304.16, 304.56, 305.27, 
306.03, 306.39, 306.86, 307.2, 307.62, 307.95, 308.49, 308.9, 
309.3, 309.69, 310.08, 310.48, 310.87, 311.27, 311.66, 312.06, 
312.45, 312.85, 313.24, 313.64, 314.03, 314.43, 314.82, 315.21, 
315.61, 316, 316.4, 316.79, 317.19, 317.58, 317.98, 318.37, 318.77, 
319.16, 319.55, 319.95, 320.34, 320.74, 321.13, 321.53, 321.92, 
322.32, 323.03, 324.13, 324.49, 325.03, 325.77, 326.1, 326.57, 
326.9, 327.43, 327.85, 328.24, 328.64, 329.03, 329.43, 329.82, 
330.22, 330.61, 331, 331.4, 331.79, 332.19, 332.58, 332.98, 333.37, 
333.79, 334.23, 334.7, 335.03, 335.55, 335.89, 336.36, 336.71, 
337.02, 337.47, 338, 338.51, 338.9, 339.3, 339.69, 340.09, 340.48, 
340.88, 341.27, 341.67, 342.06, 342.45, 342.85, 343.24, 343.64, 
344.03, 344.43, 344.82, 345.22, 345.61, 346.01, 346.4, 346.8, 
347.19, 347.58, 347.98, 348.49, 348.88, 349.24, 349.55, 349.89, 
350.32, 350.65, 350.98, 351.51, 352.25, 353.19, 353.49, 353.83, 
354.27, 354.6, 354.93, 355.46, 355.88, 356.27, 356.67, 357.06, 
357.46, 357.85, 358.25, 358.64, 359.3, 360.16, 360.78, 361.24, 
361.79, 362.27, 362.83, 363.19, 363.79)), class = "data.frame", row.names = c(NA, 
-848L))

From which i get

plot(df$time, df$Survival, ylim=c(0,1), type="s", lwd=2, bty="n", xlim=c(0,400))

How can I calculate the smoothed instantaneous hazard function from this dataframe? I dont have access to number at risk.

what exactly do you mean by 'hazard'? hazard is a function that varies over time and can only be approximated by the data you have. so you'll have to assume some kind of model or a use a smoothing function. — George Savva, Nov 14 '21 at 10:56
I would like to calculate the smooth instantanous hazard fuction from the data above. — ECII, Nov 14 '21 at 11:04
We need the number at risk parameter from the reported survival curves! Or at least we need to see the original survival curve! — TarJae, Nov 14 '21 at 13:09
do you know the total number of risk at the outset? otherwise could you assume each step represents one failure? — George Savva, Nov 14 '21 at 13:13
@TarJae the X axis is in Days. Number at risk at t=0 is 700. Patient at risk at other times are not provided. — ECII, Nov 14 '21 at 16:12
Do you now the interval on x axis say 1 5 10 15 20 or 1 100 200 300? — TarJae, Nov 14 '21 at 16:15
Ok. It is worth to see here This is the code for this instructions I tried it and it works. But what you need is the number at risk. Maybe you could estimate it by try and error comparing the final plot with the original plot. Unfortunately with my guess HR was NA. Good luck! — TarJae, Nov 14 '21 at 20:30

IRTFM · Answer 1 · 2022-01-08T20:59:32.340

The theoretical instantaneous hazard is the time derivative of the cumulative Hazard function which is just 1-Survival. I disagree that the "number at risk" should be necessary in this instance, because you have already created an empirical Survival function. Looking at the Survival curve you would expect a correct estimate to have an early peak and then taper off. So see whether this estimator suffices:

#Pseudo-code
IHF <- { ( S(x+1) - S(x) ) /S(x) }/ # the S(x) in denominator replaces  n-at-risk
            (t(x+1)-t(x)
# R-code
IHF <- function( S, t) { (-diff(S)/head(S, -1))/diff(t) }
plot( head(df$time,-1) , IHF(df$Survival, df$time) )

Well wasn't entirely satisfactory. You only have 11 instances where there is a decrement, so I guess I can see why you want a smoothed estimator. Lets see if loess can suffice. Nope. Couldn't get a sufficient bandwidth. The muhaz package is designed for this. I'm not sure if it's the best for it, but it's the one I'm most familiar with. It designed for censored data but the default for its censoring status is to assume no cases are censored, so yo can just provide times and it will take the survival into account:

fit2 <- muhaz(head(df$time,-1)[ as.logical(diff(df$Survival)) ], maxtime=360)
plot(fit2)

I wasn't entirely happy with that result, being concerned about the late peak. So I tried anotehr package:

install.packages("bshazard"
library(bshazard
times=head(df$time,-1)[ as.logical(diff(df$Survival)) ]
 times
#[1]   0.87   3.48  13.72  33.80  75.68  90.39 145.66 200.85 272.76 295.29 352.25
  fit<-bshazard( Surv(times,rep(1, length(times))) ~ 1, data=df)

I line that one a bit better. (Which is a bit ironic since I'm the maintainer of pkg:muhaz, but I'm just a caretaker.)

Calculate hazard from digitized survival curves

1 Answers1