Absolute direction of bias/indirectness
Absolute directions of bias/indirectness
The first step of the adjustment is to calculate the absolute direction of bias/indirectness. A helpful way to understand this. plotted on a standard forest plot. Note that the red arrows show the absolute direction of bias, not the direction of adjustment.
For additive bias/indirectness, the position of the point estimate does not effect the absolute direction of bias. That is, regardless of whether the point estimate is above or below the null, the absolute direction of bias/indirectness will be the same.
In contrast, for proportional bias/indirectness, the absolute direction of bias depends on the position of the effect estimate. There are two scenarios to consider:
Point estimate below NULL
If the effect estimate is below the null, then bias towards the null would be adjusted for by moving the effect estimate proportionally to the left. Conversely, bias away from the null would be adjusted for moving the point estimate proportionally to the right.
Point estimate above NULL - Bias towards the NULL
In contrast, if the effect estimate is below the null, then bias towards the null would be adjusted for by moving the effect estimate proportionally to the left. Conversely, bias away from the null would be adjusted for moving the point estimate proportionally to the right.
For example, if the effect estimate represents a protective effect (below the null), then bias towards the null would be adjusted for by moving the effect estimate proportionally to the right. In contrast, if the effect of the intervention is harmful (effect estimate above the null), bias towards the null would be adjusted for by moving the effect estimate proportionally to the left.
Adding the adjustment values
The absolute direction is also used when defining the sign of the
prior in the tri_append_*() functions. If the
bias/indirectness is expected to pull the effect to the
right (absolute direction = right), it is
given a positive sign. This is because in
tri_calculate_adjusted_estimates(), we are subtracting the
total additive bias from the effect estimate \(y_i\).
So to correct for an absolute direction to the right, we want the sign of the prior to be positive (e.g. N(0.9, 0.5)), so that when it is subtracted from the effect estimate, the overall impact is negative (and so we shift the effect estimate to the left):
\(y_i - (0.6)\) = \(y_i - 0.6\)
Conversely, when the assessment is that the absolute direction of bias is to the left, we want the sign of the bias correction to be negative (e.g. N(-0.9, 0.5)), so that we are shifting the effect estimate to the right:
\(y_i - (-0.6)\) = \(y_i + 0.6\)
See Section 6.1 of Turner et al. for the derivation of these equations.
Example
To illustrate this and ensure that our code is working correctly, lets consider a simple dataset of 6 studies, with each study fulfilling one of the scenarios described above.
# Create the example dataset
example_data <- tibble(
result_id = paste0("S", 1:6), # <-- Add this line
study = paste("Study", 1:6),
yi = c(-.2, .25, -.1, -.3, .1, .3),
vi = round(runif(6, 0.01, 0.05), 3), # use positive variances
d1j = rep("moderate", 6),
d1t = c(rep("add", 2), rep("prop", 4)),
d1d = c("Favours comparator", "Favours experimental",
"Away from null", "Towards null",
"Away from null", "Towards null")
)
# Complete the preparation steps of triangulate
example_data_prepped <- example_data %>%
tri_to_long() %>%
tri_absolute_direction() %>%
tri_append_bias(dat_bias_values)Additive - Favours comparator - right - positive sign
example_data[1,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S1 Study 1 -0.2 0.03 moderate add Favours comparator
example_data_prepped[1,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S1 Study 1 -0.2 0.03 d1 modera… add right 0.09 0.05
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>Additive - Favours intervention - left - negative sign
example_data[2,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S2 Study 2 0.25 0.031 moderate add Favours experimental
example_data_prepped[2,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S2 Study 2 0.25 0.031 d1 modera… add left -0.09 0.05
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>Proportional - Point estimate above NULL - Towards the NULL - left - negative sign
example_data[3,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S3 Study 3 -0.1 0.017 moderate prop Away from null
example_data_prepped[3,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S3 Study 3 -0.1 0.017 d1 modera… prop left 0 0
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>Proportional - Point estimate below NULL - Away from NULL - right - positive sign
example_data[4,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S4 Study 4 -0.3 0.043 moderate prop Towards null
example_data_prepped[4,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S4 Study 4 -0.3 0.043 d1 modera… prop right 0 0
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>Proportional - Point estimate below NULL - Towards the NULL - right - positive sign
example_data[5,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S5 Study 5 0.1 0.039 moderate prop Away from null
example_data_prepped[5,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S5 Study 5 0.1 0.039 d1 modera… prop right 0 0
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>Proportional - Point estimate below NULL - Away from NULL - left - negative sign
example_data[6,]
#> # A tibble: 1 × 7
#> result_id study yi vi d1j d1t d1d
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr>
#> 1 S6 Study 6 0.3 0.013 moderate prop Towards null
example_data_prepped[6,]
#> # A tibble: 1 × 12
#> result_id study yi vi domain j t d bias_m_add bias_v_add
#> <chr> <chr> <dbl> <dbl> <chr> <chr> <chr> <chr> <dbl> <dbl>
#> 1 S6 Study 6 0.3 0.013 d1 modera… prop left 0 0
#> # ℹ 2 more variables: bias_m_prop <dbl>, bias_v_prop <dbl>