I've been receiving the warning message in the title and have reviewed posts such as e.g. this one.
I would like to understand how this feature has perfect separation with the target variable, since I just assumed that this kind of warning would be more associated with categorical features, where one particular level has all of either true or false target class.
The context is website conversion (transaction makes a purchase True = X1 or not = False X0). I wanted to understand the impact of average page load time for a given website session. After removing other features like device type and traffic source, I have found that I only receive the warning with the feature Avg_Load_Time which is a numeric (dbl) feature.
My next thought was that maybe all those sessions with 0 avg load time were causing perfect separation however I have no zeros, just some close to 0:
> summary(x$Avg_Load_Time)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.24 2.32 4.27 10.18 8.73 484.62
I then looked at a summary of Avg Load Time only for those sessions with a transaction, where target is thus X1:
> summary(y %>% filter(target == "X1") %>% select(Avg_Load_Time))
Avg_Load_Time
Min. : 0.780
1st Qu.: 2.478
Median : 3.785
Mean : 4.253
3rd Qu.: 4.815
Max. :16.410
I can see here that while the min is higher, it's not 0.
How can I find the cause of my perfect separation given I've narrowed it down to a single feature?
Here is a sample of 1000 if it helps. Any tips on understanding my separation appreciated:
dput(x %>% sample_n(1000))
structure(list(target = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 1L,
2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L), .Label = c("X0", "X1"), class = "factor"), Avg_Load_Time = c(0.77,
39.1, 5.34, 5.45, 1.74, 2.18, 9.19, 4.73, 9.37, 2.45, 4.33, 1.86,
1.93, 4.32, 18.13, 6.93, 3.57, 13.93, 130.38, 4.47, 26.67, 14.48,
19.54, 9.41, 6.51, 3.78, 1.91, 2.98, 5.47, 2.24, 3.07, 27.9,
8.8, 65.66, 10.23, 3.32, 1.81, 5.02, 2.71, 1.04, 11.76, 5.73,
2.32, 3.54, 2.3, 63.9, 4.5, 0.78, 1.44, 4.06, 0.7, 1.79, 7.7,
4.3, 33.25, 1.44, 0.79, 6.39, 4.17, 0.6, 3.58, 16.84, 11.07,
16.05, 28.29, 9.22, 4.1, 7.81, 0.55, 64.88, 3.32, 10.44, 3.22,
1.57, 1.01, 7.16, 3.41, 5.74, 3.73, 2.62, 4.39, 17.92, 5.05,
1.94, 6.95, 1.86, 27.07, 7.69, 4.05, 2.96, 8.03, 3.21, 5.33,
1.62, 17.03, 8.37, 1.7, 5.08, 4.96, 0.83, 4.65, 16.36, 7.04,
4.9, 22.98, 6.08, 4.3, 2.91, 1.52, 1.81, 11.28, 16.71, 4.17,
9.62, 3.18, 2.66, 0.78, 9.3, 25.39, 5.84, 1.13, 58.03, 1.45,
10.45, 19.5, 1.25, 1.06, 30.49, 2.9, 7.31, 3.61, 4.64, 0.68,
10.43, 8.84, 1.78, 17.16, 6.68, 4.61, 7.43, 5.03, 2.98, 2.89,
4.15, 9.47, 3.68, 2.16, 2.09, 41.78, 3.06, 113.4, 30.13, 5.37,
14.83, 2.1, 2.03, 13.51, 3.1, 5.54, 4.61, 18.09, 23.82, 34.64,
4.99, 8.35, 7.45, 3.98, 3.44, 1.01, 34.45, 64.03, 2.82, 13.63,
13.34, 0.66, 4.15, 2.06, 19.7, 1.38, 2.16, 10.65, 5.89, 57.27,
17.51, 3.5, 10.97, 2.2, 9.38, 2.06, 5.25, 4.11, 72.22, 0.93,
3.65, 5.71, 4.79, 3.01, 0.95, 6.6, 15.35, 1.05, 3.31, 3.44, 8.31,
11.35, 6.63, 4.87, 4.83, 10.05, 1.01, 25.35, 3.79, 11.14, 24.26,
9.71, 1.76, 3.75, 1.66, 7.02, 6.41, 3.72, 3.58, 35.16, 3.24,
2.29, 9.61, 9.31, 0.67, 0.63, 7.08, 10.85, 2.65, 4.35, 5.86,
3.24, 4.32, 3.34, 2.37, 4.23, 1.97, 1.83, 15.42, 4.17, 5.18,
2.37, 8.91, 0.71, 20.18, 5.96, 1.41, 3.11, 26.85, 2.47, 5.99,
2.53, 1.86, 2.67, 13.66, 8.28, 5.7, 8.1, 3.95, 139.35, 15.37,
2.55, 2.85, 5.46, 2.55, 17.16, 2.87, 23.42, 1.58, 62.58, 7.5,
14.41, 1.57, 4.42, 5.41, 4.62, 12.5, 3.3, 4.37, 3.91, 3.35, 7.27,
1.11, 24.86, 18, 8.83, 7.87, 2.68, 2.77, 32.58, 12.66, 2.64,
9.89, 30.86, 10.17, 3.49, 37.99, 4.99, 12.98, 1.75, 11.92, 45.36,
3.35, 2.28, 2.83, 19.92, 9.33, 4.98, 19.76, 2.92, 3.84, 4.8,
205.98, 4.53, 8.82, 3.74, 21.8, 3.56, 3.9, 2.29, 7.85, 79.96,
3.56, 2.78, 5.9, 2.93, 3.76, 1.79, 12.94, 2.34, 25.17, 22.71,
4.15, 6.87, 147.62, 6.1, 3.23, 93.41, 12.91, 4.93, 3.22, 5.84,
8.73, 17.73, 79.63, 182.45, 2.36, 1.62, 1.22, 1.09, 3.75, 0.93,
1.82, 12.14, 4.38, 2.1, 0.88, 4.36, 1.33, 3.74, 2.85, 2.34, 13.2,
5.44, 9.94, 6.6, 2.79, 7.7, 10.99, 11.43, 19.7, 3.79, 2.26, 1.68,
23.24, 7.41, 3.13, 5.22, 2.4, 4.48, 2.35, 10.36, 1.25, 34.14,
7.37, 3.46, 18.84, 8.32, 4.9, 2.37, 1.03, 4.56, 9.7, 20.95, 1.01,
17.42, 9.29, 0.88, 3.84, 13.82, 0.52, 4.51, 11.74, 1, 6.28, 5.49,
6.13, 5.62, 0.53, 6.72, 2.08, 3.38, 68.72, 4.56, 2.45, 15.21,
5.54, 5.13, 3.86, 4.89, 1.21, 3.88, 4.83, 4.97, 8.22, 5.76, 4.07,
6.83, 1.94, 120.71, 3.26, 7.38, 4.21, 5.95, 3.7, 1.28, 3.43,
1.42, 1.63, 3.97, 10.57, 8.98, 2.37, 21.73, 8.04, 5.18, 2.48,
5.74, 4.65, 1.85, 6.75, 0.98, 1.72, 4, 6.08, 7.21, 8, 10.98,
1.94, 0.75, 30.3, 7.29, 3.31, 4.3, 66.62, 3.87, 3.01, 1.56, 3.37,
5.44, 6.76, 6.21, 1.39, 8.02, 2.95, 9.56, 1.62, 2.28, 0.46, 2,
12.55, 4.66, 15.48, 1.76, 5.81, 1.94, 4.25, 2.65, 1.51, 2.7,
27.43, 46.24, 2.67, 16.77, 0.7, 0.4, 6.07, 11.3, 1.49, 3.45,
3.2, 22.74, 1.5, 0.7, 2.6, 7.89, 2.57, 3.42, 2.46, 1.7, 2.45,
2.12, 7.97, 9.4, 3.58, 7.2, 12.18, 15.27, 2.94, 5.19, 7.33, 7.54,
5.01, 5.08, 10.65, 16.13, 2.46, 5.28, 3.02, 2.82, 10.84, 0.53,
4.22, 3.51, 10.69, 4.31, 2.55, 7.58, 19.3, 4.97, 9.39, 1.66,
0.45, 2.71, 0.82, 0.7, 8.76, 21.98, 1.95, 1.09, 3.78, 2.71, 2.55,
1.69, 17.2, 6.37, 11.42, 2.33, 0.98, 52.6, 1.67, 1.32, 21.99,
34.11, 4.99, 4.52, 6.84, 2.45, 0.7, 1.16, 9.52, 21.73, 2.32,
5.26, 7.34, 3.55, 2.6, 4.29, 9.48, 0.48, 7.22, 1.94, 4.25, 6.62,
6.76, 3.39, 1.67, 3.81, 38.39, 3.49, 65.29, 3.59, 11.54, 1.87,
4.21, 6.6, 7.3, 8.97, 9.82, 2.65, 4.99, 2.03, 4.81, 3.08, 6.41,
1.29, 1.04, 3.53, 1.29, 4.07, 2.92, 2.91, 3.82, 4.94, 2.25, 10.05,
8.87, 1.51, 3.26, 3.4, 0.68, 7.64, 0.6, 0.78, 6.25, 2.89, 17.56,
4.83, 5.55, 9.6, 3.31, 2.43, 6.96, 5.05, 5.95, 6.96, 15.06, 45.99,
1.74, 3.48, 1.83, 2.76, 6.35, 24.95, 1.96, 2.23, 2.23, 17.25,
5.2, 12.57, 11.58, 10.85, 2.91, 1.1, 3.2, 6.4, 3.15, 5.55, 1.72,
2.34, 1.83, 49.76, 1.87, 5.72, 3.59, 0.81, 8.8, 6.76, 2.06, 3.15,
9.06, 15.15, 1.64, 4.92, 9.64, 3.7, 1.78, 1.88, 3.98, 4.93, 3.37,
10.57, 4.41, 4.67, 6.39, 3.51, 21.83, 2.33, 0.68, 1.66, 2.89,
4.57, 360.7, 5.89, 6.63, 8.59, 0.48, 8.08, 2.01, 1.59, 12.45,
0.99, 2.3, 2.79, 1.47, 2.78, 2.05, 3.12, 17.84, 185.53, 3.71,
0.8, 1.82, 12.42, 31.16, 2.27, 19.23, 1.48, 7.22, 0.24, 11.73,
1.25, 14.06, 11.55, 1.48, 1.73, 5.01, 1.66, 2.25, 3.26, 6.73,
4.66, 1.8, 5.25, 8.15, 3.94, 2.72, 1.69, 25.96, 4.46, 1.51, 1.61,
1.67, 2.16, 5.24, 22.86, 3.64, 10.68, 4.65, 0.62, 0.64, 7.69,
3.63, 37.52, 9.98, 3.27, 10.94, 1.92, 2.4, 1.04, 6.05, 5.34,
3.4, 4.08, 72.08, 3.95, 5.1, 1.44, 17.06, 2.14, 4.17, 3.39, 7.79,
5.71, 19.87, 2.54, 2.49, 3.44, 3.85, 12.06, 12.18, 1.7, 3.12,
17.3, 4.41, 4.4, 0.82, 57.91, 124.91, 5.35, 5.41, 20.75, 13.54,
0.82, 0.84, 8.62, 10.04, 1.08, 10.49, 7.05, 2.72, 1.18, 2.05,
6.87, 3.51, 20.66, 4.69, 31.9, 4.64, 6.04, 1.71, 6.91, 70.11,
2.83, 9.88, 2, 10.48, 4.25, 12.24, 1.27, 50.22, 0.85, 3.51, 5.47,
0.69, 1.45, 2.97, 1.58, 2.2, 6.79, 15.88, 3.52, 1.75, 18.68,
3.81, 2.87, 4.06, 69.44, 91.15, 0.79, 1.15, 6.57, 1.18, 4.33,
7.3, 42.46, 40.83, 6.48, 32.34, 3.16, 41.11, 4.61, 1.57, 2.22,
1.2, 2.35, 10.48, 6.82, 5.38, 5.51, 3.34, 57.3, 51.9, 10.52,
1.85, 3.37, 4.42, 1.09, 29.53, 1.76, 2.48, 2.54, 10.22, 11.62,
59.79, 176.17, 7.18, 4.36, 1.76, 7.34, 4.55, 8.21, 3.94, 9.64,
1.62, 19.5, 5.53, 5.28, 1.59, 43.85, 24.02, 5.95, 6.34, 4.54,
3.71, 1.48, 9.18, 5.56, 6.08, 15.67, 24.48, 0.8, 12.53, 4.14,
29.11, 19.85, 2.54, 92.42, 44.65, 8.07, 2.44, 3.93, 3.79, 13.65,
17.64, 3.67, 9.42, 3.43, 1.81, 11.76, 1.63, 4.27, 5.87, 11.66,
3.77, 1.62, 3.58, 15.66, 4.46, 8.12, 7.35, 8.62, 6.24, 4.28,
1.68, 3.93, 3.27, 2.67, 2.93, 161.22, 3.54, 2.62, 40.6, 1.09,
2.3, 9.57, 1.1, 3.33, 17.41, 7.63, 4.01, 16.9, 3.8, 2.8, 3.56,
2.51, 6.26, 1.84, 2.98, 4.92, 2.12, 6.35, 11.74, 2.64, 14.35,
452.01, 1.7, 1.91, 4.79, 2.49, 7.61, 1.54, 8.19, 7.95, 2.81,
7.08, 9.06, 5.17, 2.08, 7.92, 4.39, 22.12, 3.42, 3.82, 3.17,
17.41, 3.29, 10.66, 31.54, 3.62, 26.38, 3.43, 10.32, 1.32, 10.71,
2.75, 0.95)), row.names = c(6184L, 2551L, 2196L, 1039L, 2202L,
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Edit:
Here's the full code I'm suing to runt he model:
library(caret)
## custom evaluation metric function
my_summary <- function(data, lev = NULL, model = NULL){
a1 <- defaultSummary(data, lev, model)
b1 <- twoClassSummary(data, lev, model)
c1 <- prSummary(data, lev, model)
out <- c(a1, b1, c1)
out}
## tuning & parameters
set.seed(123)
train_control <- trainControl(
method = "cv",
number = 5,
savePredictions = TRUE,
verboseIter = TRUE,
classProbs = TRUE,
summaryFunction = my_summary
)
linear_model = train(
x = select(training_data, Avg_Load_Time),
y = target,
trControl = train_control,
method = "glm", # logistic regression
family = "binomial",
metric = "AUC"
)
After running this, I get the warning message.
Best Answer
I looked at your data and it is extremely skewed with outliers. Thus you do not have perfect separation but the warning is occurring because some of the extreme observations have predicted probabilities indistinguishable from 1.
If you fit the model on the log of avg_load_time you will not get the error (I tested this on your sample data).
This answer explains what's going on well: Issue with complete separation in logistic regression (in R)