Compute Standardized Mean Difference

Computes the standardized mean differnce (SMD) between two groups.

$$ d = \sqrt{D' S^{-1} D} $$

where \(D\) is a vector of differences between group 1 and 2 and \(S\) is the covariance matrix of these differences. If \(D\) is length 1, the result is multplied by \(sign(D)\).

In the case of a numeric or integer variable, this is equivalent to:

$$ d = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{(s^2_1 + s^2_2)/2}} $$ where \(\bar{x}_g\) is the sample mean for group \(g\) and \(s^2_g\) is the sample variance.

For a logical or factor with only two levels, the equation above is \(\bar{x}_g = \hat{p}_g\), i.e. the sample proportion and \(s^2_g = \hat{p}_g(1 - \hat{p}_g)\).

When using the SMD to evaluate the effectiveness of weighting in achieving covariate balance, it is important to isolate the change in SMD before and after weighting to the change in mean difference, so the denominator (covariance matrix) must be held constant (Stuart 2008, doi:10.1002/sim.3207 ). By default, the unweighted covariance matrix is used to compute SMD in both the unweighted and weighted case. If the weights are not being used to adjust for covariate imbalance (e.g. case weights), the unwgt.var argument can be set to FALSE to use the weighted covariance matrix as the denominator.

smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for character,ANY,missing
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for character,ANY,numeric
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for logical,ANY,missing
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for logical,ANY,numeric
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for matrix,ANY,missing
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for matrix,ANY,numeric
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for list,ANY,missing
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for list,ANY,numeric
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for data.frame,ANY,missing
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

# S4 method for data.frame,ANY,numeric
smd(x, g, w, std.error = FALSE, na.rm = FALSE, gref = 1L, unwgt.var = TRUE)

Arguments

x

a vector or matrix of values

g

a vector of at least 2 groups to compare. This should coercable to a factor.

w

a vector of numeric weights (optional)

std.error

Logical indicator for computing standard errors using compute_smd_var. Defaults to FALSE.

na.rm

Remove NA values from x? Defaults to FALSE.

gref

an integer indicating which level of g to use as the reference group. Defaults to 1.

unwgt.var

Use unweighted or weighted covariance matrix. Defaults to TRUE

Value

a data.frame containing standardized mean differences between levels of g for values of x. The data.frame contains the columns:

• term: the level being comparing to the reference level

• estimate: SMD estimates

• std.error: (if std.error = TRUE) SMD standard error estimates

Examples

x <- rnorm(100)
g <- rep(1:2, each = 50)
smd(x, g)
#>   term   estimate
#> 1    2 -0.3213335