# Preface

I was basically done with this blog post when I came across Matti Vuorre’s post on the same exact topic. Matti goes into *all* the details, and really the present post can be seen as a brief account of all the cool things the probit-approach-to-SDT can do. I’m only posting this here because I really like my plots 🤷

Previously, we’ve seen that for data from a binary decision signal detection task, we can use a *probit binomial* regression model (like a logistic regression, but with a probit link function) to estimate the two main parameters of *signal detection theory* (SDT): the sensitivity and the bias.

In this post I would like to show how this idea can be extended to multiple response SDT tasks by using an *ordinal probit* regression model.

# The Data

Imagine the following task: after being presented with 20 images of dogs, you are presented with 300 new images of dogs, and you have to decide for each dog if it appeared in the training set (“Old”) or not (“New”).

In a binary decision task, you would simply indicate “New” or “Old”, but in this task you have multiple response options - from 1 to 6, with 1 = “Feels New” and 6 = “Feels Old”. We can call this scale a “feelings numbers” scale.

After going over all 30 photos, you have

`STD_data`

```
## # A tibble: 12 × 3
## Truth Response N
## <fct> <ord> <dbl>
## 1 New Confidence1 35
## 2 New Confidence2 31
## 3 New Confidence3 26
## 4 New Confidence4 22
## 5 New Confidence5 19
## 6 New Confidence6 17
## 7 Old Confidence1 14
## 8 Old Confidence2 20
## 9 Old Confidence3 22
## 10 Old Confidence4 27
## 11 Old Confidence5 32
## 12 Old Confidence6 35
```

Where *N* is the number of responses in each condition and response level.

# Modeling with Classic SDT

We can use Siegfried Macho’s R code to extract the SDT parameters. In this case, they are:

*Sensitivity*- The distance between the two (latent) normal distributions. The further they are, the more “distinguishable” the Old and New images are from each other.

*5 Threshold*- One between each pair of consecutive possible responses. Perceived “stimulation” above each threshold leads to a decision in that category.

(These will probably make sense when we present them visually below.)

First, we’ll model this with classical SDT:

```
SDT_equal <- SDT.Estimate(
data = STD_data[["N"]],
test = TRUE,
# We have 2 option: Old / New; We'll assume equal variance
n = list(n.sdt = 2, restriction = "equalvar")
)
SDT.Statistics(SDT_equal)[["Free.parameters"]]
```

```
## Value SE CFI-95(Lower) CFI-95(Upper)
## Mean[2] 0.564 0.040 0.486 0.642
## t-1 -0.744 0.034 -0.810 -0.678
## t-2 -0.165 0.031 -0.226 -0.104
## t-3 0.267 0.031 0.206 0.329
## t-4 0.707 0.033 0.643 0.772
## t-5 1.260 0.036 1.189 1.331
```

# Modeling as a Probit Cumulative Ordinal

```
library(dplyr) # 1.0.9
library(tidyr) # 1.2.0
library(ordinal) # 2019.12.10
library(parameters) # 0.18.2
library(ggplot2) # 3.3.6
library(patchwork) # 1.1.1
```

We can also model this data with a Probit Cumulative Ordinal model, predicting the *Response* from the *Truth*:
- The slope of *Truth* indicates the effect of the true image identity had on the response pattern - this is ** sensitivity**.

- In an ordinal model, we get

*k*-1 “intercepts” (

*k*being the number of unique responses). Each intercept represents the value above which a predicted value will be binned into the next class. There represent our

**.**

*shreshold*```
m_equal <- clm(Response ~ Truth,
data = STD_data,
weights = N,
link = "probit"
)
parameters::model_parameters(m_equal) |>
insight::print_html()
```

Model Summary | |||||
---|---|---|---|---|---|

Parameter | Coefficient | SE | 95% CI | z | p |

Intercept | |||||

Confidence1|Confidence2 | -0.74 | 0.10 | (-0.94, -0.54) | -7.21 | < .001 |

Confidence2|Confidence3 | -0.16 | 0.10 | (-0.35, 0.02) | -1.72 | 0.085 |

Confidence3|Confidence4 | 0.27 | 0.10 | (0.08, 0.46) | 2.81 | 0.005 |

Confidence4|Confidence5 | 0.71 | 0.10 | (0.51, 0.90) | 7.08 | < .001 |

Confidence5|Confidence6 | 1.26 | 0.11 | (1.05, 1.48) | 11.38 | < .001 |

Location Parameters | |||||

Truth (Old) | 0.57 | 0.12 | (0.33, 0.81) | 4.65 | < .001 |

As we can see, the estimated values are identical!^{1}

The advantage of the probit ordinal model is that it is easy(er) to build this model up:

- Add predictors of sensitivity (interactions with
`Truth`

)

- Add predictors of bias (main effects / intercept effects)

- Add random effects (with
`ordinal::clmm()`

)

(*See Matti’s post for actual examples!*)

```
mean2 <- coef(m_equal)[6]
Thresholds <- coef(m_equal)[1:5]
ggplot() +
# Noise
stat_function(aes(linetype = "Noise"), fun = dnorm,
size = 1) +
# Noise + Signal
stat_function(aes(linetype = "Noise + Signal"), fun = dnorm,
args = list(mean = mean2),
size = 1) +
# Thresholds
geom_vline(aes(xintercept = Thresholds, color = names(Thresholds)),
size = 2) +
scale_color_brewer("Threshold", type = "div", palette = 2,
labels = paste0(1:5, " | ", 2:6)) +
labs(y = NULL, linetype = NULL, x = "Obs. signal") +
expand_limits(x = c(-3, 3), y = 0.45) +
theme_classic()
```

# Unequal Variance

The standard model of SDT assumes that the *Noise* and the *Noise + Signal* distribution differ only in their mean; that is, *N+S* is a shifted *N* distribution. This is almost always not true, with \(\sigma_{\text{N+S}}>\sigma_{\text{N}}\).

To deal with this, we can also estimate the variance of the *N+S* distribution.

First, with the classic SDT model:

```
SDT_unequal <- SDT.Estimate(
data = STD_data[["N"]],
test = TRUE,
# We have 2 option: Old / New; Not assuming equal variance
n = list(n.sdt = 2, restriction = "no")
)
SDT.Statistics(SDT_unequal)[["Free.parameters"]]
```

```
## Value SE CFI-95(Lower) CFI-95(Upper)
## Mean[2] 0.552 0.041 0.473 0.632
## Stddev[2] 0.960 0.035 0.891 1.029
## t-1 -0.728 0.036 -0.799 -0.658
## t-2 -0.159 0.032 -0.221 -0.097
## t-3 0.266 0.031 0.205 0.327
## t-4 0.696 0.034 0.629 0.763
## t-5 1.235 0.043 1.151 1.318
```

And with a probit ordinal regression, but allow the latent scale to vary:

```
m_unequal <- clm(Response ~ Truth,
scale = ~ Truth, # We indicate that the scale is a function of the underlying dist
data = STD_data,
weights = N,
link = "probit"
)
parameters::model_parameters(m_unequal) |>
insight::print_html()
```

Model Summary | |||||
---|---|---|---|---|---|

Parameter | Coefficient | SE | 95% CI | z | p |

Intercept | |||||

Confidence1|Confidence2 | -0.72 | 0.11 | (-0.93, -0.50) | -6.51 | < .001 |

Confidence2|Confidence3 | -0.16 | 0.10 | (-0.34, 0.03) | -1.61 | 0.107 |

Confidence3|Confidence4 | 0.27 | 0.10 | (0.08, 0.45) | 2.80 | 0.005 |

Confidence4|Confidence5 | 0.69 | 0.10 | (0.49, 0.90) | 6.66 | < .001 |

Confidence5|Confidence6 | 1.23 | 0.13 | (0.98, 1.49) | 9.50 | < .001 |

Location Parameters | |||||

Truth (Old) | 0.55 | 0.12 | (0.31, 0.80) | 4.49 | < .001 |

Scale Parameters | |||||

Truth (Old) | -0.05 | 0.12 | (0.31, 0.80) | 4.49 | < .001 |

The scale parameter needs to be back transformed to get the sd of the *N+S* distribution: \(e^{-0.05}=0.95\), and so one again the estimated values are identical!

```
mean2 <- coef(m_unequal)[6]
sd2 <- exp(coef(m_unequal)[7])
Thresholds <- coef(m_unequal)[1:5]
ggplot() +
# Noise
stat_function(aes(linetype = "Noise"), fun = dnorm,
size = 1) +
# Noise + Signal
stat_function(aes(linetype = "Noise + Signal"), fun = dnorm,
args = list(mean = mean2, sd = sd2),
size = 1) +
# Thresholds
geom_vline(aes(xintercept = Thresholds, color = names(Thresholds)),
size = 2) +
scale_color_brewer("Threshold", type = "div", palette = 2,
labels = paste0(1:5, " | ", 2:6)) +
labs(y = NULL, linetype = NULL, x = "Obs. signal") +
expand_limits(x = c(-3, 3), y = 0.45) +
theme_classic()
```

## ROC Curve or ROC Curve*s*?

An additional check we can preform is whether the various responses are indeed the product of single ROC curve. We do this by plotting the ROC curve on a inv-normal transformation (that is, converting probabilities into normal quantiles). Quantiles that fall on a straight line indicate they are part of the same curve.

```
pred_table <- data.frame(Truth = c("Old", "New")) |>
mutate(predict(m_unequal, newdata = cur_data(), type = "prob")[[1]] |> as.data.frame()) |>
tidyr::pivot_longer(starts_with("Confidence"), names_to = "Response") |>
tidyr::pivot_wider(names_from = Truth)
ROC_table <- pred_table |>
rows_append(data.frame(New = 0, Old = 0)) |>
mutate(
Sensitivity = lag(cumsum(New), default = 0),
Specificity = rev(cumsum(rev(Old))),
)
p_roc <- ggplot(ROC_table, aes(Sensitivity, Specificity)) +
geom_line() +
geom_abline(slope = 1, intercept = 1, linetype = "dashed") +
geom_point(aes(color = ordered(Response)), size = 2) +
expand_limits(x = c(0,1), y = c(0,1)) +
scale_x_continuous(trans = "reverse") +
scale_color_brewer("Threshold", type = "div", palette = 2,
labels = paste0(1:5, " | ", 2:6),
na.translate = FALSE) +
labs(color = NULL) +
theme_classic()
p_zroc <- ROC_table |>
tidyr::drop_na(Response) |>
ggplot(aes(qnorm(Sensitivity), qnorm(Specificity))) +
geom_line() +
geom_point(aes(color = ordered(Response)), size = 2) +
expand_limits(x = c(0,1), y = c(0,1)) +
scale_x_continuous(trans = "reverse") +
scale_color_brewer("Threshold", type = "div", palette = 2,
labels = paste0(1:5, " | ", 2:6),
na.translate = FALSE) +
labs(color = NULL, x = "Z(Sensitivity)", y = "Z(Specificity)") +
theme_classic()
p_roc + p_zroc + plot_layout(guides = "collect")
```

`## Warning: Removed 1 rows containing missing values (geom_point).`

## Going Full Bayesian

Although the `clmm()`

function allows for multilevel probit regression, it does not^{2} support varying scale parameter.

Alas, we *must* use `brms`

.

```
library(brms) # 2.17.0
library(ggdist) # 3.1.1
library(tidybayes) # 3.0.2
```

In `brms`

we will use the `cumulative()`

family, which has a family-parameter called `disc`

which gives the standard deviation of the latent distributions. I will set some weak priors on the mean and standard deviation of the *N+S* distribution, and I will also set the standard deviation of the *N* distribution to 1 (on a log scale, to 0) using the `constant(0)`

prior.

```
b_formula <- bf(Response | weights(N) ~ Truth,
disc ~ 0 + Intercept + Truth)
b_priors <-
set_prior("normal(0, 3)", coef = "TruthOld") +
set_prior("normal(0, 1.5)", coef = "TruthOld", dpar = "disc") +
set_prior("constant(0)", coef = "Intercept", dpar = "disc")
Bayes_mod <- brm(b_formula,
prior = b_priors,
family = cumulative(link = "probit", link_disc = "log"),
data = STD_data,
backend = "cmdstanr",
refresh = 0
)
```

```
model_parameters(Bayes_mod, test = NULL) |>
insight::print_html()
```

Model Summary | ||||
---|---|---|---|---|

Parameter | Median | 95% CI | Rhat | ESS |

Intercept[1] | -0.73 | (-0.95, -0.52) | 1.002 | 3141.00 |

Intercept[2] | -0.16 | (-0.35, 0.03) | 1.000 | 4737.00 |

Intercept[3] | 0.27 | (0.09, 0.46) | 1.000 | 5290.00 |

Intercept[4] | 0.71 | (0.51, 0.91) | 1.000 | 3911.00 |

Intercept[5] | 1.26 | (1.01, 1.52) | 1.000 | 2833.00 |

TruthOld | 0.57 | (0.33, 0.81) | 1.000 | 3735.00 |

disc_Intercept | 0.00 | (0.00, 0.00) | ||

disc_TruthOld | 0.02 | (-0.20, 0.24) | 1.001 | 2394.00 |

```
criteria <- gather_rvars(Bayes_mod, Intercept[Response])
signal_dist <- spread_draws(Bayes_mod, b_TruthOld, b_disc_TruthOld) |>
mutate(b_disc_TruthOld = exp(b_disc_TruthOld)) |>
group_by(.draw) |>
summarise(
x = seq(-3, 3, length = 20),
d = dnorm(x, mean = b_TruthOld, b_disc_TruthOld)
) |>
ungroup() |>
curve_interval(.along = x, .width = 0.9)
ggplot() +
# Noise
stat_function(aes(linetype = "Noise"), fun = dnorm) +
# Noise + Signal
geom_ribbon(aes(x = x, ymin = .lower, ymax = .upper),
data = signal_dist,
fill = "grey", alpha = 0.4) +
geom_line(aes(x, d, linetype = "Noise + Signal"), data = signal_dist) +
# Thresholds
stat_slab(aes(xdist = .value, fill = ordered(Response)),
color = "gray", alpha = 0.6, key_glyph = "polygon",
data = criteria) +
# Theme and scales
scale_fill_brewer("Threshold", type = "div", palette = 2,
labels = paste0(1:5, " | ", 2:6),
na.translate = FALSE) +
labs(color = NULL, linetype = NULL, x = "Obs. signal", y = NULL) +
theme_classic()
```

This gives roughly the same results as `clm()`

, but would allow for multilevel modeling of both the location and scale of the latent variable.