Chapter 4 Linear Models
##
## Attaching package: 'dplyr'## The following objects are masked from 'package:stats':
##
## filter, lag## The following objects are masked from 'package:base':
##
## intersect, setdiff, setequal, union4.1 Figures
Why is the normal so normal?
## simulate people stepping
thousand_steppers <- replicate(1e3, runif(16, -1, 1), simplify = FALSE)
steppers_df <- thousand_steppers %>%
map(cumsum) %>%
map_df(~ data_frame(position = c(0, .x),
step_num = seq_along(position)),
.id = "stepper")
steppers_df %>%
ggplot(aes(x = step_num, y = position, group = stepper)) +
geom_line(alpha = 0.2)
steppers_df %>%
filter(step_num == 16) %>%
ggplot(aes(x = position)) +
geom_density() +
stat_function(fun = dnorm, colour = "green", args = list(sd = 2.25))
Many small products can also cause a normal distribution
replicate(1e3, prod(1 + runif(12, 0, 0.1))) %>%
density() %>%
plot
## these parameter values ar just eyeballed
curve(dnorm(x, mean = 1.8, sd = 0.2), add = TRUE, col = "green")
4.1.1 Gaussian model of height
start by digging out the data.
data("Howell1", package = "rethinking")
knitr::kable(head(Howell1))| height | weight | age | male |
|---|---|---|---|
| 151.765 | 47.82561 | 63 | 1 |
| 139.700 | 36.48581 | 63 | 0 |
| 136.525 | 31.86484 | 65 | 0 |
| 156.845 | 53.04191 | 41 | 1 |
| 145.415 | 41.27687 | 51 | 0 |
| 163.830 | 62.99259 | 35 | 1 |
## we're working only with adults
d2 <- Howell1 %>% filter(age > 18)It is a good idea to plot one’s priors
data_frame(x = c(100, 250)) %>%
ggplot(aes(x = x)) +
stat_function(fun = dnorm, args = list(mean = 178, sd = 20))
data_frame(x = c(0, 60)) %>%
ggplot(aes(x = x)) +
stat_function(fun = dunif, args = list(min = 0, max = 50))
remember, every posterior is also potentially a prior for a subsequent analysis
Sounds like an important quote! but how do you use empirical priors?
Anyways, We are recommended to try to combine these priors in a simulation, to “see what these priors imply about the distribution of individual heights”. This despite what is said later in the book, about how priors role is “epistemilogical, not mechanistic” (or something like that).
rnorm(
n = 1e4,
mean = rnorm(1e4, 178, 20),
sd = runif(1e4, 0, 50)
) %>%
data_frame(priors = .) %>%
ggplot(aes(x = priors)) + geom_density()
4.1.2 grid approximation
post <- expand.grid(
mu = seq(from = 140, to = 160, length.out = 200),
sigma = seq(from = 4 , to = 9 , length.out = 200)
) %>%
mutate(LL = map2_dbl(mu, sigma,
~ d2$height %>%
dnorm(mean = .x, sd = .y, log = TRUE) %>%
sum))
## that gives us the liklihood. next we multiply by the prior -- or, since this
## is a log scale, we add it to the prior
height_posterior <- post %>%
mutate(prod = LL +
dnorm(mu, 178, 20, log = TRUE) +
dunif(sigma, 0, 50, log = TRUE),
prob = exp(prod - max(prod)))
## why on earth do we subtract the maximum??with(height_posterior,
rethinking::contour_xyz(mu, sigma, prob))
We can also use sample_n() to create a resampled posterior fairly easily?
height_posterior$prob %>% sum ## er should this be 1???## [1] 308.6014height_post_sampled <- height_posterior %>%
sample_n(1e4, replace = TRUE, weight = prob)
height_post_sampled %>%
ggplot(aes(x = mu, y = sigma)) +
stat_bin_hex(aes(fill = ..density..), bins = 30)## Loading required package: methods
long_height_post <- height_post_sampled %>%
select(mu, sigma) %>%
tidyr::gather(param, value)
long_height_post %>%
ggplot(aes(x = value)) + geom_density() + facet_grid(~ param, scales = "free_x") + geom_rug() 
Analyzing data from only 20 values of height:
## 20 subsamples
twenty_heights <- d2 %>% sample_n(20) %>% .[["height"]]
post <- tidyr::crossing(
mu.list = seq(from = 120, to = 170, length.out = 200),
sigma.list = seq(from = 4, to = 20, length.out = 200)
) %>%
mutate(LL = map2_dbl(mu.list, sigma.list,
~ dnorm(
x = twenty_heights,
mean = .x, sd = .y,
log = TRUE) %>%
sum))
post %>%
mutate(prod = LL +
dnorm(mu.list, 178, 20, TRUE) +
dunif(sigma.list, 0, 50, TRUE),
prob = exp(prod - max(prod))) %>%
## sample posterior
sample_n(1e4, weight = prob) %>%
ggplot(aes(x = mu.list, y = sigma.list)) +
stat_bin_hex(aes(fill = ..density..), bins = 37)
4.1.3 Fittin the model with MAP
data("Howell1", package = "rethinking")
## we're working only with adults
d2 <- Howell1 %>% filter(age > 18)
flist <- alist(
height ~ dnorm(mu, sigma),
mu ~ dnorm(178, 20),
sigma ~ dunif(0, 50)
)
m4.1 <- rethinking::map(flist, data = d2)
rethinking::precis(m4.1)## An object of class "precis"
## Slot "output":
## Mean StdDev 5.5% 94.5%
## mu 154.654530 0.4172152 153.987740 155.321321
## sigma 7.762328 0.2950798 7.290733 8.233922
##
## Slot "digits":
## [1] 2WE could also get the HDPI for the marginal posterior
m4.1 %>%
rethinking::extract.samples(n = 1e4) %>%
map(rethinking::HPDI)## $mu
## |0.89 0.89|
## 153.9675 155.2916
##
## $sigma
## |0.89 0.89|
## 7.243285 8.204885Note that the estimates are conditional on each others priors. For example:
m4.2 <- rethinking::map(
alist(
height ~ dnorm(mu, sigma),
mu ~ dnorm(178, 0.1),
sigma ~ dunif(0, 50)
),
data = d2)
rethinking::precis(m4.2)## An object of class "precis"
## Slot "output":
## Mean StdDev 5.5% 94.5%
## mu 177.86598 0.1002314 177.70579 178.0262
## sigma 24.48462 0.9356075 22.98934 25.9799
##
## Slot "digits":
## [1] 2To simulate confidence intervals that include the covariation, we simulate from a bivariate normal distribution:
rethinking::vcov(m4.1)## mu sigma
## mu 0.1740685496 0.0002278744
## sigma 0.0002278744 0.08707211004.2 Adding a predictor
m4.3 <- rethinking::map(
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * weight,
a ~ dnorm(156, 100),
b ~ dnorm(0, 10),
sigma ~ dunif(0, 50)
),
data = d2
)
rethinking::precis(m4.3, corr = TRUE)## An object of class "precis"
## Slot "output":
## Mean StdDev 5.5% 94.5% a
## a 113.8239752 1.93872846 110.725513 116.9224378 1.0000000000
## b 0.9062098 0.04260545 0.838118 0.9743015 -0.9899123065
## sigma 5.1093962 0.19423065 4.798978 5.4198143 0.0006398681
## b sigma
## a -0.9899123065 0.0006398681
## b 1.0000000000 -0.0006337993
## sigma -0.0006337993 1.0000000000
##
## Slot "digits":
## [1] 2These parameters are highly correlated.
## same model as before with centered variables
d2 %>%
mutate(weight = weight - mean(weight)) %>%
rethinking::map(data = .,
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * weight,
a ~ dnorm(156, 100),
b ~ dnorm(0, 10),
sigma ~ dunif(0, 50)
)
) %>%
rethinking::precis(., corr = TRUE)## An object of class "precis"
## Slot "output":
## Mean StdDev 5.5% 94.5% a
## a 154.6443791 0.27468127 154.2053853 155.0833728 1.000000e+00
## b 0.9065486 0.04261321 0.8384445 0.9746528 5.819321e-10
## sigma 5.1093863 0.19422970 4.7989697 5.4198028 2.841610e-06
## b sigma
## a 5.819321e-10 2.841610e-06
## b 1.000000e+00 -2.935764e-05
## sigma -2.935764e-05 1.000000e+00
##
## Slot "digits":
## [1] 2Now to plot the predictions
d2 %>%
ggplot(aes(x = weight, y = height)) +
geom_point() +
geom_abline(intercept = rethinking::coef(m4.3)["a"],
slope = rethinking::coef(m4.3)["b"])
The correlated values of the posterior distribution contain the slope and intercept of many lines:
m4.3 <- d2 %>%
rethinking::map(data = .,
alist(
height ~ dnorm(mu, sigma),
mu <- a + b * weight,
a ~ dnorm(156, 100),
b ~ dnorm(0, 10),
sigma ~ dunif(0, 50)
)
)
d2 %>%
ggplot(aes(x = weight, y = height)) +
geom_point() +
geom_abline(aes(intercept = a,
slope = b),
data = rethinking::extract.samples(m4.3, n = 20),
alpha = 0.6)
what is the predicted height when weight is 50kg?
rethinking::extract.samples(m4.3, n = 300) %>%
mutate(mean_height_50 = a + b * 50) %>%
ggplot(aes(x = mean_height_50)) + geom_density() + geom_rug(alpha = 0.6)
This is the error about one point on the line. What about over the whole line?
## there is a function in rethinking that will create a matrix of predicted values
mu <- rethinking::link(m4.3)## [ 100 / 1000 ]
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[ 1000 / 1000 ]dim(mu)## [1] 1000 346nrow(d2)## [1] 346Sure enough, there are as many columns as there were observations in the input data – and as many rows as there are samples from the posterior.
## create a data.frame for reorganizing
long_predictions <- mu %>%
as.data.frame %>%
mutate(sim_number = seq_along(V1)) %>%
tidyr::gather(observation_id, datapoint, -sim_number)
intervals <- long_predictions %>%
group_by(observation_id) %>%
tidyr::nest() %>%
mutate(rangedata = map(data, ~ rethinking::HPDI(.x$datapoint)),
lower = map_dbl(rangedata, 1),
upper = map_dbl(rangedata, 2)) %>%
select(observation_id, lower, upper)
d2 %>%
bind_cols(intervals) %>%
ggplot(aes(x = weight, y = height, ymax = upper, ymin = lower)) + geom_pointrange()
Not quite what we want! better would be to have predictions for a range of original values:
library(tidyr)
library(modelr)
prediction_range <- d2 %>%
data_grid(weight = seq_range(weight, 50))
model_grid_predictions <- prediction_range %>%
rethinking::link(m4.3, data = .) %>%
as.data.frame %>%
mutate(sim_number = seq_along(V1)) %>%
gather(observation_id, datapoint, -sim_number) %>%
nest(-observation_id) %>%
bind_cols(prediction_range)## [ 100 / 1000 ]
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[ 1000 / 1000 ]We can visualize the distribution of the posterior at every value of x this way:
library(ggforce)
model_grid_predictions %>%
unnest(data) %>%
ggplot(aes(x = weight, y = datapoint, group = observation_id)) +
geom_sina(alpha = 0.01, pch = 21, fill = "black")And we can add that uncertainty to the line itself as well:
mean_and_raw <- model_grid_predictions %>%
mutate(rangedata = map(data, ~ rethinking::HPDI(.x$datapoint)),
lower = map_dbl(rangedata, 1),
upper = map_dbl(rangedata, 2),
mean = map_dbl(data, ~ mean(.$datapoint))) %>%
select(observation_id, weight, lower, upper) %>%
ggplot(aes(x = weight, ymax = upper, ymin = lower)) +
geom_ribbon() +
geom_point(aes(x = weight, y = height, ymax = NULL, ymin = NULL), data = d2)
mean_and_raw
Figure 4.1: prediction of the mean as a function of weight, plotted with actual data as well.
simulated_data <- prediction_range %>%
rethinking::sim(m4.3, data = .) %>%
as.data.frame %>%
mutate(sim_number = seq_along(V1)) %>%
gather(observation_id, datapoint, -sim_number) %>%
nest(-observation_id) %>%
bind_cols(prediction_range)## [ 100 / 1000 ]
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[ 1000 / 1000 ]simulated_data %>% head## observation_id
## 1 V1
## 2 V2
## 3 V3
## 4 V4
## 5 V5
## 6 V6
## data
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146.8759, 137.7974, 143.5189, 141.6742, 139.4152, 153.7638, 152.4492, 144.4708, 141.9637, 140.1371, 141.6323, 146.0993, 143.4075, 141.9255, 149.3382, 158.8001, 148.3604, 147.0785, 141.4802, 146.0317, 154.1263, 148.3054, 141.0191, 154.6868, 145.0805, 146.1788, 150.6459, 141.5645, 149.6842, 143.6881, 147.7190, 145.2333, 140.9966, 142.1832, 146.6237, 146.3338, 155.3134, 134.2471, 145.7552, 142.4514, 146.8528, 141.5577, 146.4119, 140.9686, 144.1959, 140.6765, 143.0905, 145.8005, 136.2491, 144.8534, 141.6630, 139.7470, 155.0872, 147.2679, 147.8913, 145.8034, 145.5171, 149.8949, 140.8720, 143.3990, 134.0419, 139.8161, 146.8272, 138.7760, 160.3123, 145.3596, 133.2346, 155.1892, 144.1801, 140.9122, 144.0349, 143.7494, 142.2945, 141.1076, 149.5013, 142.2654, 132.5825, 154.9055, 141.5944, 143.7894, 147.6657, 143.9848, 143.9719, 146.8754, 147.9955, 145.7392, 145.0320, 146.1854, 140.9999, 142.5722, 153.5602, 141.9539, 136.1500, 147.6219, 152.2607, 138.8088, 151.5411, 140.2529, 142.0535, 143.6802, 147.8265, 137.1505, 142.8188, 151.1297, 153.2265, 142.2025, 146.0712, 146.4631, 152.3519, 145.4166, 140.0296, 144.7993, 146.2356, 146.1625, 149.1659, 153.9082, 152.6123, 142.1476, 145.1590, 144.8425, 145.2464, 141.1116, 153.4976, 139.7764, 139.2346, 146.9896, 141.9590, 139.6905, 147.0407, 139.9073, 137.2738, 139.9209, 149.0689, 145.6026, 148.0428, 139.2436, 132.7312, 143.3323, 151.9575, 146.7131, 141.6004, 141.1550, 145.8288, 151.4936, 143.9029, 144.3286, 150.8122, 149.4514, 146.4344, 153.4291, 145.4931, 145.4466, 144.3738, 142.7681, 140.1439, 140.5057, 141.6219, 158.1146, 149.9076, 150.2318, 144.3981, 140.6013, 148.7388, 135.7357, 139.7153, 151.5123, 150.0146, 142.0450, 148.6705, 142.5089, 154.5468, 138.4155, 144.9230, 145.6855, 145.0959, 147.3346, 135.3178, 137.4402, 149.1021, 147.4493, 140.2160, 154.7542, 141.9157, 153.2852, 136.4347, 130.2152, 148.8374, 151.7479, 147.5884, 143.1028, 144.2782, 154.1358, 143.2865, 144.6926, 145.2211, 146.3398, 153.2241, 152.1331, 140.5027, 146.2519, 158.2209, 151.4296, 151.4490, 138.5522, 145.5087, 147.9335, 142.9364, 142.0526, 138.7001, 144.2593, 140.2048, 147.4236, 143.8228, 142.5887, 139.4140, 153.6117, 139.4769, 143.5515, 140.9130, 144.2786, 156.0728, 144.0211, 145.0898, 140.8840, 149.1947, 140.9709, 148.6098, 146.5683, 138.3983, 140.0312, 152.4370, 143.5585, 137.6832, 144.8056, 146.0188, 148.7614, 146.5906, 141.2113, 140.9216, 139.3053, 143.2013, 138.4818, 139.1475, 156.0985, 143.4030, 148.3929, 146.9293, 149.0457, 144.1069, 149.0662, 146.7311, 155.5274, 143.9750, 146.3296, 153.8731, 138.2383, 145.9385, 141.6218, 140.2705, 137.5085, 138.7912, 138.7214, 152.9469, 145.1356, 149.9930, 145.1180, 151.0236, 150.9094, 148.0797, 140.1558, 148.2896, 136.1902, 138.8886, 147.7694, 146.4382, 147.3303, 144.4425, 143.5971, 146.8207, 140.8569, 149.7611, 141.9525, 143.0558, 141.6531, 143.6694, 145.0044, 139.1931, 142.4264, 135.9011, 148.7584, 141.8537, 149.0061, 130.7719, 147.8415, 153.8854, 145.3748, 146.4599, 145.7362, 147.2397, 137.4133, 147.1797, 139.3212, 148.6754, 149.8592, 149.2200, 141.8820, 148.8747, 138.4357, 142.9449, 141.9964, 147.0569, 143.1344, 144.6737, 149.5005, 142.3908, 147.4045, 147.6231, 148.4890, 145.2328, 142.5143, 152.4852, 147.1313, 137.7240, 144.9173, 137.0426, 139.5127, 153.9295, 139.1767, 146.8740, 142.3339, 148.8998, 143.3842, 153.4307, 152.2038, 142.9198, 146.4988, 145.2465, 145.2113, 144.1499, 151.0123, 139.5420, 143.7946, 139.0529, 151.1922, 140.2477, 141.8439, 142.6068, 145.3776, 140.6269, 137.8642, 148.1615, 153.3271, 141.5825, 147.6531, 144.7144, 139.0153, 142.5440, 145.4327, 143.0489, 149.2052, 151.6464, 143.2656, 143.6192, 143.3043, 149.1364, 135.5900, 145.5325, 149.6919, 142.7523, 145.0812, 144.4601, 143.0780, 155.3549, 138.2346, 142.7220, 152.3054, 148.7519, 151.5617, 151.7743, 148.4129, 142.5765, 142.5899, 144.5638, 153.9358, 139.1579, 138.2550, 143.8407, 139.5322, 142.7494, 157.0553, 141.5594, 140.6576, 135.1488, 141.6886, 138.5856, 151.8226, 148.3941, 142.2830, 139.9202, 140.4276, 131.5973, 148.2331, 139.2621, 141.7019, 151.4724, 147.0282, 138.7982, 141.2669, 147.7220, 148.4786, 146.6227, 141.8170, 142.7570, 150.7175, 145.2013, 141.5374, 141.0010, 152.3430, 144.7848, 141.6004, 149.7511, 154.7496, 145.4636, 150.4252, 143.0816, 147.3563, 145.0296, 137.7276, 152.2254, 137.1996, 153.3015, 146.9067, 138.7101, 145.8230, 143.7605, 142.3583, 138.6569, 140.4381, 156.6222, 145.7915, 152.9961, 139.4508, 141.8648, 142.3212, 140.9460, 149.4291, 148.9349, 137.4945, 138.6384, 137.5208, 148.2937, 151.8938, 150.1443, 133.1620, 146.4903, 143.0612, 152.4832, 156.2790, 142.9445, 148.0093, 147.6667, 145.8304, 148.2204, 140.7266, 153.3419, 150.8767, 146.8195, 143.0795, 149.9793, 148.8358, 140.6145, 143.6915, 148.1379, 147.5178, 150.6210, 143.6823, 147.2399, 148.9613, 141.1809, 148.8473, 144.6788, 131.7982, 141.3081, 149.6534, 139.7106, 149.2390, 145.8838, 145.8496, 150.1856, 142.1623, 146.0658, 147.2568, 143.7798, 148.9203, 143.9928, 147.9042, 136.3841, 139.1156, 140.1522, 148.0526, 148.6863, 145.6104, 148.4075, 146.0076, 148.1655, 154.1550, 147.9750, 142.3603, 146.4315, 148.5751, 143.9472, 149.7508, 144.2183, 141.8989, 152.0695, 147.4455, 144.2136, 142.1667, 142.2458, 141.8029, 149.3032, 137.4011, 129.6686, 149.0500, 146.4549, 142.9334, 147.6236, 152.3617, 141.8811, 142.8320, 155.5227, 139.9891, 150.9237, 138.1863, 146.9016, 146.9052, 146.7411, 149.2336, 137.6210, 137.2784, 137.1808, 149.9635, 146.7915, 142.6326, 150.0983, 145.8509, 146.7973, 138.5621, 145.3525, 143.1193, 136.3077, 146.3355, 145.5296, 156.5166, 148.4016, 139.3758, 146.6624, 146.5491, 152.3991, 156.6149, 149.7739, 152.2456, 148.5446, 151.2793, 150.8866, 141.9120, 151.7050, 142.6356, 153.2049, 147.1380, 144.8459, 149.6471, 140.5908, 145.5933, 143.7974, 137.9190, 149.3516, 148.5779, 137.9088, 135.8471, 146.0372, 156.4116, 146.9514, 144.0711, 144.7633, 141.7980, 146.9825, 145.9448, 151.9771, 145.9369
## weight
## 1 31.52464
## 2 32.16685
## 3 32.80905
## 4 33.45125
## 5 34.09346
## 6 34.73566prediction_range <- simulated_data %>%
mutate(height_pi = map(data, ~ rethinking::PI(.x$datapoint)),
height_pi = map(height_pi,
~ data_frame(lower_pi = .x[[1]],
upper_pi = .x[[2]]))) %>%
unnest(height_pi)
mean_and_raw +
geom_ribbon(aes(ymin = lower_pi, ymax= upper_pi), data = prediction_range, alpha = 0.2) +
theme_minimal()
Figure 4.2: predicting the actual data. this includes the predicted standard deviation of the response.
4.2.1 Polynomial
d <- Howell1
str(d)## 'data.frame': 544 obs. of 4 variables:
## $ height: num 152 140 137 157 145 ...
## $ weight: num 47.8 36.5 31.9 53 41.3 ...
## $ age : num 63 63 65 41 51 35 32 27 19 54 ...
## $ male : int 1 0 0 1 0 1 0 1 0 1 ...d_stand_weight <- d %>%
mutate(weight.s = (weight - mean(weight)) / sd(weight))
d_stand_weight %>%
select(height, starts_with("weight")) %>%
gather(wtvar, weight, starts_with("weight")) %>%
ggplot(aes(x = weight, y = height)) + geom_point() + facet_grid(~wtvar, scales = "free_x")
quad_mod <- alist(
height ~ dnorm(mu, sigma),
mu <- a + b1 * weight.s + b2 * weight.s2,
a ~ dnorm(178, 100),
b1 ~ dnorm(0, 10),
b2 ~ dnorm(0, 10),
sigma ~ dunif(0, 50)
)
m4.5 <- d_stand_weight %>%
mutate(weight.s2 = weight.s^2) %>%
rethinking::map(flist = quad_mod,
data = .)
rethinking::precis(m4.5)## Mean StdDev 5.5% 94.5%
## a 146.66 0.37 146.07 147.26
## b1 21.40 0.29 20.94 21.86
## b2 -8.42 0.28 -8.86 -7.97
## sigma 5.75 0.17 5.47 6.034.3 Exercises
4.3.1 easy
4.3.2 medium
4.3.3 hard
4.4 When adding variables hurts
# number of plants
N <- 100
set.seed(4812)
fungal_data <- data_frame(
# initial heights
h0 = rnorm(N, 10, 2),
treatment = rep(0:1, each = N/2),
fungus = rbinom(N, size = 1, prob = 0.5 - treatment*0.4),
h1 = h0 + rnorm(N, 5 - 3 * fungus),
# departure from the book, but why not center h0??
h0_c = h0 - mean(h0)
)fungal_data %>%
ggplot(aes(x = h0_c, y = h1, fill = factor(fungus))) + geom_point(pch = 21) + facet_wrap(~treatment) + stat_smooth(method = "lm")
4.4.1 the trouble with fitting post-treatment effects
Sometimes biologists are tempted to include both a treatment and one of the responses in a linear model.
m5.13_define <- alist(
h1 ~ dnorm(mu, sigma),
mu <- a + b_trt * treatment + b_fungus * fungus + b_h * h0_c,
a ~ dnorm(0, 100),
c(b_trt, b_fungus, b_h) ~ dnorm(0, 10),
sigma ~ dunif(0, 10)
)
m5.13 <- rethinking::map(m5.13_define, data = as.data.frame(fungal_data))
rethinking::precis(m5.13)## Mean StdDev 5.5% 94.5%
## a 15.05 0.18 14.76 15.35
## b_trt 0.07 0.21 -0.27 0.41
## b_fungus -3.19 0.23 -3.55 -2.83
## b_h 1.00 0.04 0.93 1.07
## sigma 0.94 0.07 0.84 1.05alright so that can’t be the right answer!! One solution, proposed at this point in the book, is to instead remove the fungal treatment from the model.
m5.14_define <- alist(
h1 ~ dnorm(mu, sigma),
mu <- a + b_trt * treatment + b_h * h0_c,
a ~ dnorm(0, 100),
c(b_trt, b_h) ~ dnorm(0, 10),
sigma ~ dunif(0, 10)
)
m5.14 <- rethinking::map(m5.14_define, data = as.data.frame(fungal_data))
rethinking::precis(m5.14)## Mean StdDev 5.5% 94.5%
## a 13.27 0.23 12.90 13.64
## b_trt 1.47 0.33 0.95 1.99
## b_h 1.04 0.08 0.92 1.16
## sigma 1.64 0.12 1.45 1.82Now we see that plants definitely grow better with the treatment!
4.4.2 Could multilevel models help here?
This is not the point at this juncture in the book, but I am curious!
m5.13_hier_def <- alist(
h1 ~ dnorm(mu, sigma),
mu <- b0 + b_h * h0_c + b_fungus * fungus,
fungus ~ dbinom(1, ppp),
logit(ppp) <- b0f + b_trt * treatment,
c(b0, b_h, b_fungus) ~ dnorm(0, 5),
c(b0f, b_trt) ~ dnorm(0, 5),
sigma <- dcauchy(0,1)
)
m5.13_hier <- rethinking::map2stan(m5.13_hier_def, data = as.data.frame(fungal_data))
saveRDS(m5.13_hier, "stan_models/m5.13_hier.rds")m5.13_hier <- readRDS("stan_models/m5.13_hier.rds")
# what are the coefficients:
rethinking::precis(m5.13_hier)## Mean StdDev lower 0.89 upper 0.89 n_eff Rhat
## b0 15.09 0.12 14.91 15.28 1000 1
## b_h 1.00 0.04 0.93 1.07 1000 1
## b_fungus -3.22 0.20 -3.53 -2.92 858 1
## b0f 0.24 0.28 -0.20 0.71 1000 1
## b_trt -2.27 0.50 -3.07 -1.55 1000 1
## sigma 0.96 0.07 0.85 1.06 802 1# fit to fake data for predictions
df <- data_frame(treatment = rep(c(0, 1), each = 10),
h0_c = rep(seq(from = -5, to = 5, length.out = 10), 2))
hier_predictions <- rethinking::link(m5.13_hier, data = as.data.frame(df), flatten = FALSE)## [ 100 / 1000 ]
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[ 1000 / 1000 ]hier_predictions$ppp %>% t %>% cbind(df, .) %>%
select(-h0_c) %>%
group_by(treatment) %>%
tidyr::nest() %>%
mutate(interval = purrr::map(data, ~ .x %>% as.matrix %>% rethinking::HPDI(.)),
low = purrr::map_dbl(interval, 1),
high = purrr::map_dbl(interval, 2))## # A tibble: 2 x 5
## treatment data interval low high
## <dbl> <list> <list> <dbl> <dbl>
## 1 0 <tibble [10 x 1,000]> <dbl [2]> 0.58293263 0.6163986
## 2 1 <tibble [10 x 1,000]> <dbl [2]> 0.05744481 0.2019443hier_predictions$mu %>% apply(., 2, rethinking::HPDI) %>% t %>% cbind(df, .)## treatment h0_c |0.89 0.89|
## 1 0 -5.0000000 6.432998 7.319180
## 2 0 -3.8888889 7.593195 8.350001
## 3 0 -2.7777778 8.748718 9.396839
## 4 0 -1.6666667 9.956380 10.522566
## 5 0 -0.5555556 11.083284 11.577541
## 6 0 0.5555556 12.178865 12.662983
## 7 0 1.6666667 13.264975 13.807678
## 8 0 2.7777778 14.315762 14.936999
## 9 0 3.8888889 15.421004 16.153823
## 10 0 5.0000000 16.452730 17.298147
## 11 1 -5.0000000 6.432998 7.319180
## 12 1 -3.8888889 7.593195 8.350001
## 13 1 -2.7777778 8.748718 9.396839
## 14 1 -1.6666667 9.956380 10.522566
## 15 1 -0.5555556 11.083284 11.577541
## 16 1 0.5555556 12.178865 12.662983
## 17 1 1.6666667 13.264975 13.807678
## 18 1 2.7777778 14.315762 14.936999
## 19 1 3.8888889 15.421004 16.153823
## 20 1 5.0000000 16.452730 17.298147# link gives me the output of each linear model in the equation. The linear models give me the mean growth and the mean probability of fungal infection. Each linear model only uses one independent variable -- the model for the probability of fungus infection uses treatment, and the model for the expected growth uses the starting size (h0_c)Suppose I want to re-create my figure, to see how the model thinks this system ought to behave (important step for model checking):
# start with samples from the posterior
m5.13_hier_samples <- rethinking::extract.samples(m5.13_hier)
m5.13_hier_samples %>% purrr::map(head)## $b0
## [1] 15.07409 14.73151 15.10655 15.07113 15.18437 15.06501
##
## $b_h
## [1] 0.9736490 0.9876098 0.9821342 1.0393464 1.0600321 1.0016488
##
## $b_fungus
## [1] -3.179489 -3.097534 -3.405055 -3.226120 -3.176906 -3.172974
##
## $b0f
## [1] 0.33482385 0.47428983 0.28648295 -0.26575485 0.36274726 -0.06148448
##
## $b_trt
## [1] -3.132594 -1.848476 -2.721778 -1.598002 -1.994654 -1.747220
##
## $sigma
## [1] 0.9236528 0.8890455 1.0085262 0.8885583 0.8820609 0.9462573as.data.frame(m5.13_hier_samples) %>% head## b0 b_h b_fungus b0f b_trt sigma
## 1 15.07409 0.9736490 -3.179489 0.33482385 -3.132594 0.9236528
## 2 14.73151 0.9876098 -3.097534 0.47428983 -1.848476 0.8890455
## 3 15.10655 0.9821342 -3.405055 0.28648295 -2.721778 1.0085262
## 4 15.07113 1.0393464 -3.226120 -0.26575485 -1.598002 0.8885583
## 5 15.18437 1.0600321 -3.176906 0.36274726 -1.994654 0.8820609
## 6 15.06501 1.0016488 -3.172974 -0.06148448 -1.747220 0.9462573samples_df <- as.data.frame(m5.13_hier_samples)
# function which creates a sampled data_frame
simulate_fungus_data <- function(b0, b_h, b_fungus, b0f, b_trt, sigma){
df %>%
mutate(prob_fungus = rethinking::inv_logit( b0f + b_trt * treatment),
has_fungus = rbinom(length(prob_fungus), size = 1, prob = prob_fungus),
mean_growth = b0 + b_h * h0_c + b_fungus * has_fungus,
sim_growth = rnorm(length(mean_growth), mean_growth, sigma))
}
fake_data <- samples_df %>%
purrr::pmap(simulate_fungus_data) %>%
bind_rows(.id = "sim")
gg_fake <- fake_data %>%
ggplot(aes(x = h0_c, y = sim_growth)) +
geom_hex(bins = 9) + facet_grid(has_fungus~treatment)
gg_fake +
geom_point(aes(x = h0_c, y = h1, color = factor(has_fungus)), pch = 21, data = fungal_data %>% rename(has_fungus = fungus))
How about some nice summary stats of this?
hier_predictions <- rethinking::link(m5.13_hier,
data = data.frame(treatment = c(0, 1),
h0_c = c(0)),
replace = list(b_fungus = matrix(1.5, nrow = 1000, ncol = 2)))## [ 100 / 1000 ]
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[ 1000 / 1000 ]hier_predictions$mu %>% head## [,1] [,2]
## [1,] 16.57409 16.57409
## [2,] 16.23151 16.23151
## [3,] 16.60655 16.60655
## [4,] 16.57113 16.57113
## [5,] 16.68437 16.68437
## [6,] 16.56501 16.56501rethinking::stancode(m5.13_hier)## data{
## int<lower=1> N;
## int<lower=1> N_fungus;
## real h1[N];
## int fungus[N_fungus];
## real h0_c[N];
## int treatment[N_fungus];
## }
## parameters{
## real b0;
## real b_h;
## real b_fungus;
## real b0f;
## real b_trt;
## real<lower=0> sigma;
## }
## model{
## vector[N_fungus] ppp;
## vector[N] mu;
## sigma ~ cauchy( 0 , 1 );
## b_trt ~ normal( 0 , 5 );
## b0f ~ normal( 0 , 5 );
## b_fungus ~ normal( 0 , 5 );
## b_h ~ normal( 0 , 5 );
## b0 ~ normal( 0 , 5 );
## for ( i in 1:N_fungus ) {
## ppp[i] = b0f + b_trt * treatment[i];
## }
## fungus ~ binomial_logit( 1 , ppp );
## for ( i in 1:N ) {
## mu[i] = b0 + b_h * h0_c[i] + b_fungus * fungus[i];
## }
## h1 ~ normal( mu , sigma );
## }
## generated quantities{
## vector[N_fungus] ppp;
## vector[N] mu;
## real dev;
## dev = 0;
## for ( i in 1:N_fungus ) {
## ppp[i] = b0f + b_trt * treatment[i];
## }
## dev = dev + (-2)*binomial_logit_lpmf( fungus | 1 , ppp );
## for ( i in 1:N ) {
## mu[i] = b0 + b_h * h0_c[i] + b_fungus * fungus[i];
## }
## dev = dev + (-2)*normal_lpdf( h1 | mu , sigma );
## }fungal_reduction <- fake_data %>%
select(sim, treatment, prob_fungus) %>%
distinct %>%
tidyr::spread(treatment, prob_fungus) %>%
mutate(reduction = `0` - `1`) %>%
.[["reduction"]]
rethinking::HPDI(fungal_reduction)## |0.89 0.89|
## 0.3261092 0.5773543median(fungal_reduction)## [1] 0.4367353You’ll have about 43% less fungus.
how_much_fungus_hurts <- fake_data %>%
select(sim, has_fungus, sim_growth) %>%
group_by(sim, has_fungus) %>%
summarize(meangrowth = mean(sim_growth)) %>%
tidyr::spread(has_fungus, meangrowth) %>%
mutate(reduction = (`0` - `1`)/`0`) %>%
.[["reduction"]] %>%
purrr::discard(is.na)
rethinking::HPDI(how_much_fungus_hurts)## |0.89 0.89|
## 0.04991973 0.35492453median(how_much_fungus_hurts)## [1] 0.2117909This fungus makes your crops about 20% smaller
The benefit of applying fertilizer is going to depend on how dangerous this fungus is.
samples_df %>% head## b0 b_h b_fungus b0f b_trt sigma
## 1 15.07409 0.9736490 -3.179489 0.33482385 -3.132594 0.9236528
## 2 14.73151 0.9876098 -3.097534 0.47428983 -1.848476 0.8890455
## 3 15.10655 0.9821342 -3.405055 0.28648295 -2.721778 1.0085262
## 4 15.07113 1.0393464 -3.226120 -0.26575485 -1.598002 0.8885583
## 5 15.18437 1.0600321 -3.176906 0.36274726 -1.994654 0.8820609
## 6 15.06501 1.0016488 -3.172974 -0.06148448 -1.747220 0.9462573vary_fungal_effect <- function(fungal_effect) {
samples_df %>%
mutate(b_fungus = fungal_effect) %>%
purrr::pmap(simulate_fungus_data) %>%
bind_rows(.id = "sim") %>%
group_by(sim, treatment) %>%
summarize(mean_growth_final = mean(sim_growth)) %>%
tidyr::spread(treatment, mean_growth_final) %>%
mutate(reduction = (`0` - `1`)/`0`) %>%
.[["reduction"]]
}
fungi_effects <- seq(from = 0, to = -5, by = -1) %>%
purrr::map(vary_fungal_effect)
hdpi <- function(x){
rethinking::HPDI(samples = x)
}
fungi_effects %>%
purrr::map(~ purrr::invoke_map(list(median = median, hdpi), x = .x)) %>%
purrr::map(purrr::flatten) %>% purrr::transpose() %>%
purrr::map(purrr::flatten_dbl) %>%
tibble::as_data_frame(.) %>%
cbind(fungal_effect = seq(from = 0, to = -5, by = -1), .) %>%
ggplot(aes(x = fungal_effect, ymin = `|0.89`, ymax = `0.89|`, y = median)) + geom_pointrange()
4.5 Binomial
4.6 Poisson
library(tidyverse)## Loading tidyverse: ggplot2
## Loading tidyverse: tibble
## Loading tidyverse: tidyr
## Loading tidyverse: readr
## Loading tidyverse: purrr
## Loading tidyverse: dplyr## Conflicts with tidy packages ----------------------------------------------## filter(): dplyr, stats
## lag(): dplyr, statsdata("Kline", package = "rethinking")
head(Kline)## culture population contact total_tools mean_TU
## 1 Malekula 1100 low 13 3.2
## 2 Tikopia 1500 low 22 4.7
## 3 Santa Cruz 3600 low 24 4.0
## 4 Yap 4791 high 43 5.0
## 5 Lau Fiji 7400 high 33 5.0
## 6 Trobriand 8000 high 19 4.0d <- Kline %>%
mutate(log_pop = log(population),
contact_high = if_else(contact == "high", true = 1, false = 0))poisson_tools <- alist(
total_tools ~ dpois(lambda),
log(lambda) <-
)