tom.palmer@bristol.ac.uk

This short vignette demonstrates the use of the
**bpbounds** package. This is a R implementation of the of
the nonparametric bounds for the average causal effect of Balke and Pearl (1997) and some extensions.
Currently this R package is a port of our **bpbounds**
Stata package (Palmer et al. (2011)). The
code implements the approach of calculating the bounds outlined by Ramsahai (2007), Ramsahai
(2008), and Ramsahai (2012).

We start by loading our package and the others needed for the code in this vignette.

Currently the package has one function, `bpbounds()`

,
which can accommodate the four scenarios implemented in our Stata
command. These are to calculate the bounds for a binary outcome, a
binary treatment/phenotype, and:

- a binary instrumental variable;
- a 3 category instrumental variable, e.g. a genotype in Mendelian randomization;

Bounds for these scenarios can be calculated with either

- trivariate data (in which all three variables are measured in one dataset);
- or bivariate data (in which the treatment/phenotype and instrument are measured in one dataset and the outcome and instrument are measured in another sample).

This example is taken from Table 1 of Balke and Pearl (1997). It is from a study of Vitamin A supplementation consisting of children in 450 villages. Of these, 221 villages were assigned to control and 229 to the treatment. The outcome was mortality.

First we setup a `data.frame`

of cell counts and convert
this into a `table`

. Here we follow the notation that
`x`

is the received treatment, `y`

is the outcome,
and `z`

is the randomized treatment (the instrumental
variable).

```
tab1dat <- data.frame(
z = c(0, 0, 1, 1, 1, 1, 0, 0),
x = c(0, 0, 0, 0, 1, 1, 1, 1),
y = c(0, 1, 0, 1, 0, 1, 0, 1),
freq = c(74, 11514, 34, 2385, 12, 9663, 0, 0)
)
tab1inddat <- uncount(tab1dat, freq)
xt <- xtabs(~ x + y + z, data = tab1inddat)
xt
#> , , z = 0
#>
#> y
#> x 0 1
#> 0 74 11514
#> 1 0 0
#>
#> , , z = 1
#>
#> y
#> x 0 1
#> 0 34 2385
#> 1 12 9663
```

Next we use `prop.table()`

to calculate the conditional
probabilities, \(P(Y,X|Z)\), and then
run `bpbounds()`

assuming the data is trivariate. Although we
could call `bpbounds()`

using the table of cell counts
`xt`

directly, i.e. `bpbounds(xt)`

.

```
p <- prop.table(xt, margin = 3)
p
#> , , z = 0
#>
#> y
#> x 0 1
#> 0 0.0063859165 0.9936140835
#> 1 0.0000000000 0.0000000000
#>
#> , , z = 1
#>
#> y
#> x 0 1
#> 0 0.0028113114 0.1972052257
#> 1 0.0009922276 0.7989912353
```

```
bpres <- bpbounds(p)
sbp <- summary(bpres)
print(sbp)
#>
#> Data: trivariate
#> Instrument categories: 2
#>
#> Instrumental inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1946228 0.005393689
#> P(Y|do(X=0)) 0.9936141 0.993614084
#> P(Y|do(X=1)) 0.7989912 0.999007772
#> CRR 0.8041263 1.005428354
#>
#> Monotonicity inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1946228 0.005393689
#> P(Y|do(X=0)) 0.9936141 0.993614084
#> P(Y|do(X=1)) 0.7989912 0.999007772
#> CRR 0.8041263 1.005428354
```

Therefore, the average causal effect is bounded between -0.1946 and 0.0054. The estimate of the ACE is found as \[\text{ACE} = \frac{\text{cov}(Y,Z)}{\text{cov}(X,Z)}\] as follows.

```
covyz <- cov(tab1inddat$y, tab1inddat$z)
covxz <- cov(tab1inddat$x, tab1inddat$z)
ace <- covyz / covxz
ace
#> [1] 0.003228039
```

If you already know the conditional probabilities you could pass them
to `bpbounds()`

as follows.

```
condprob <- c(.0064, 0, .9936, 0, .0028, .001, .1972, .799)
tabp <- array(condprob,
dim = c(2, 2, 2),
dimnames = list(
x = c(0, 1),
y = c(0, 1),
z = c(0, 1)
)) %>%
as.table()
bpbounds(tabp)
#>
#> Data: trivariate
#> Instrument categories: 2
#>
#> Instrumental inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1946000 0.005400
#> P(Y|do(X=0)) 0.9936000 0.993600
#> P(Y|do(X=1)) 0.7990000 0.999000
#> CRR 0.8041465 1.005435
#>
#> Monotonicity inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1946000 0.005400
#> P(Y|do(X=0)) 0.9936000 0.993600
#> P(Y|do(X=1)) 0.7990000 0.999000
#> CRR 0.8041465 1.005435
```

To demonstrate the features of the command we can treat this data as bivariate.

```
gtab <- xtabs( ~ y + z, data = tab1inddat)
gp <- prop.table(gtab, margin = 2)
gp
#> z
#> y 0 1
#> 0 0.006385916 0.003803539
#> 1 0.993614084 0.996196461
```

```
ttab <- xtabs( ~ x + z, data = tab1inddat)
tp <- prop.table(ttab, margin = 2)
tp
#> z
#> x 0 1
#> 0 1.0000000 0.2000165
#> 1 0.0000000 0.7999835
```

```
bpres2 <- bpbounds(p = gp, t = tp, fmt = "bivariate")
sbp2 <- summary(bpres2)
print(sbp2)
#>
#> Data: bivariate
#> Instrument categories: 2
#>
#> Instrumental inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1974342 0.006385916
#> P(Y|do(X=0)) 0.9936141 0.993614084
#> P(Y|do(X=1)) 0.7961799 1.196212998
#> CRR 0.8012969 1.203901009
#>
#> Monotonicity inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.1974342 0.006385916
#> P(Y|do(X=0)) 0.9936141 0.993614084
#> P(Y|do(X=1)) 0.7961799 1.002582378
#> CRR 0.8012969 1.009025933
```

Mendelian randomization is an approach in epidemiology due to Davey Smith and Ebrahim (2003) in which genotypes established to be robustly associated with phenotypes are used as instrumental variables in order to better estimate the causal effect of the phenotype on a disease outcome.

This example uses data from Meleady et al. (2003). It is trivariate data with 3 category instrument and binary phenotype and outcomes. The instrument is the 677CT polymorphism (rs1801133) in the Methylenetetrahydrofolate Reductase gene, involved in folate metabolism, as an instrumental variable to investigate the effect of homocysteine on cardiovascular disease.

The data are presented to us as conditional probabilities, so we take care to enter them in the correct position in the vectors.

```
mt3 <- c(.83, .05, .11, .01, .88, .06, .05, .01, .72, .05, .20, .03)
p3 = array(mt3,
dim = c(2, 2, 3),
dimnames = list(
x = c(0, 1),
y = c(0, 1),
z = c(0, 1, 2)
))
p3 <- as.table(p3)
bpres3 <- bpbounds(p3)
sbp3 <- summary(bpres3)
print(sbp3)
#>
#> Data: trivariate
#> Instrument categories: 3
#>
#> Instrumental inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.09 0.74000
#> P(Y|do(X=0)) 0.06 0.12000
#> P(Y|do(X=1)) 0.03 0.80000
#> CRR 0.25 13.33333
#>
#> Monotonicity inequality: FALSE
```

We see that in this example the monotonicity inequality is not satisfied.

This example recreates that found in section 8.3 of our Stata Journal
paper which uses simulated data to show that the IV inequality does not
always detect violations to the underlying model assumptions. We
simulate two outcomes, `y1`

and `y2`

, for which
the assumptions are violated because there is a direct effect of the
instrument, `z`

, on the outcome. However, the direct effect
on `y2`

is much smaller in magnitude than the direct effect
on `y1`

.

```
set.seed(2232011)
n <- 10000
z <- rbinom(n, 1, .5)
u <- rbinom(n, 1, .5)
px <- .05 + .1 * z + .1 * u
x <- rbinom(n, 1, px)
p1 <- .1 + .2 * z + .05 * x + .1 * u
y1 <- rbinom(n, 1, p1)
p2 <- .1 + .05 * z + .05 * x + .1 * u
y2 <- rbinom(n, 1, p2)
tab1 <- xtabs(~ x + y1 + z)
p1 <- prop.table(tab1, margin = 3)
bpres1 <- bpbounds(p1)
sbp1 <- summary(bpres1)
print(sbp1)
#>
#> Data: trivariate
#> Instrument categories: 2
#>
#> Instrumental inequality: FALSE
#>
#> Monotonicity inequality: FALSE
```

Due to the strong direct effect of the instrument on the outcome, both the instrumental variable and monotonicity inequalities are not satisifed.

```
tab2 <- xtabs(~ x + y2 + z)
p2 <- prop.table(tab2, margin = 3)
bpres2 <- bpbounds(p2)
sbp2 <- summary(bpres2)
print(sbp2)
#>
#> Data: trivariate
#> Instrument categories: 2
#>
#> Instrumental inequality: TRUE
#> Causal parameter Lower bound Upper bound
#> ACE -0.18711965 0.6700688
#> P(Y|do(X=0)) 0.17159525 0.2384172
#> P(Y|do(X=1)) 0.05129753 0.8416640
#> CRR 0.21515868 4.9049378
#>
#> Monotonicity inequality: FALSE
```

However, with the weaker direct effect we see that both are satisfied when the true underlying data generating model does not satisfy the instrumental variable assumptions.

In conclusion, we have demonstrated the use of our
**bpbounds** package implementing nonparametric bounds for
the average causal effect for a range of data scenarios.

Balke, A., and J. Pearl. 1997. “Bounds on
treatment effects from studies with imperfect compliance.”
*Journal of the American Statistical Association* 92 (439):
1172–76.

Davey Smith, G., and S. Ebrahim. 2003. “‘Mendelian randomization’: can genetic
epidemiology contribute to understanding environmental determinants of
disease.” *International Journal of Epidemiology*
32 (1): 1–22. https://doi.org/10.1093/ije/dyg070.

Meleady, Raymond, Per M Ueland, Henk Blom, Alexander S Whitehead, Helga
Refsum, Leslie E Daly, Stein Emil Vollset, et al. 2003.
“Thermolabile Methylenetetrahydrofolate Reductase, Homocysteine,
and Cardiovascular Disease Risk: The European Concerted Action
Project.” *The American Journal of Clinical Nutrition* 77
(1): 63–70.

Palmer, T. M., R. R. Ramsahai, V. Didelez, and N. A Sheehan. 2011.
“Nonparametric Bounds for the Causal Effect in a Binary
Instrumental-Variable Model.” *Stata Journal* 11 (3):
345–67. https://www.stata-journal.com/article.html?article=st0232.

Ramsahai, R. R. 2007. “Causal Bounds and Instruments.” In
*Proceedings of the Twenty-Third Annual Conference on Uncertainty in
Artificial Intelligence (UAI-07)*, 310–17. Corvallis, Oregon: AUAI
Press.

———. 2008. “Causal Inference with Instruments and Other
Supplementary Variables.” PhD thesis, Oxford, UK: Department of
Statistics, University of Oxford. https://ora.ox.ac.uk/objects/ora:2803.

———. 2012. “Causal Bounds and Observable Constraints for
Non-Deterministic Models.” *Journal of Machine Learning
Research* 13: 829–48.