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Approximate Case Influence Using Scores and Casewise Likelihood

2024-04-08

This vignette explains the computational shortcut implemented in semfindr to approximate the casewise influence. The approximation is most beneficial when the sample size N is large, where the approximation works better and the computational cost for refitting models N times is high.

Using Scores to Approximate Case Influence

lavaan provides the handy lavScores() function to evaluate \[s_i(\theta_m) = \frac{\partial \ell_i(\theta_m)}{\partial \theta_m}\] for observation \(i\), where \(\ell_i(\theta)\) denotes the casewise loglikelihood function and \(\theta_m\) is the \(m\)th model parameter.

For example,

head(lavScores(fit)[ , 1, drop = FALSE])
#>           m1~iv1
#> [1,]  0.23975993
#> [2,]  0.06630118
#> [3,] -0.33999129
#> [4,] -0.22536120
#> [5,]  0.61375747
#> [6,]  0.03747179

indicates the partial derivative of the casewise loglikelihod with respect to the parameter m1~iv1. Because the sum of the partial derivatives across all observations is zero at the maximum likelihood estimate with the full sample (\(\hat \theta_m\); i.e., the derivative of loglikelihood of the full data is 0), \(- s_i(\theta_m)\) can be used as an estimate of the partial derivative of the loglikelihood at \(\hat \theta_m\) for the sample without observation \(i\). This information can be used to approximate the maximum likelihood estimate for \(\theta_m\) when case \(i\) is dropped, denoted as \(\hat \theta_{m(-i)}\)

The second-order Taylor series expansion can be used to approximate the parameter vector estimate with an observation deleted, \(\hat \theta_{(-i)}\), as in the iterative Newton’s method. Specifically,

\[\hat \theta_{(i)} \approx \hat \theta - \frac{N}{N - 1}V(\hat \theta) \nabla \ell(\hat \theta)\] \[\hat \theta - \hat \theta_{(i)} \approx \frac{N}{N - 1}V(\hat \theta) \nabla \ell_i(\hat \theta),\] where \(\nabla \ell_i(\hat \theta)\) is the gradient vector of the casewise loglikelihood with respect to the parameters (i.e., score). The \(N / (N - 1)\) term is used to adjust for the decrease in sample size (this adjustment is trivial in large samples). This procedure should be the same as equation (4) of Tanaka et al. (1991) (p. 3807) and is related to the one-step approximation described by Cook and Weisberg (1982) (p. 182).

Comparison

The approximation is implemented in the est_change_raw_approx() function:

fit_est_change_approx <- est_change_raw_approx(fit)
fit_est_change_approx
#> 
#> -- Approximate Case Influence on Parameter Estimates --
#> 
#>    id m1~iv1  id m1~iv2 id  dv~m1  id m1~~m1 id dv~~dv
#> 1  51  0.042  43 -0.025 65  0.037  61  0.042 16  0.106
#> 2  43 -0.040  94  0.023 11 -0.027  85  0.040  9  0.050
#> 3  34 -0.032 100 -0.022 16 -0.024 100  0.037 76  0.049
#> 4  18 -0.028  85  0.021 32 -0.020  18  0.031 25  0.049
#> 5  13  0.028  20  0.020 99  0.020  42  0.028 91  0.043
#> 6  32 -0.025  32  0.019 79  0.018  43  0.025 17  0.039
#> 7  20 -0.024  65  0.019 93  0.018  32  0.023 26  0.028
#> 8  75  0.021  34 -0.018 22  0.017  34  0.022 65  0.027
#> 9  42 -0.020  64 -0.016 61 -0.016  40  0.022 62  0.027
#> 10 68  0.018  52  0.016 25 -0.015  20  0.022 90  0.024
#> 
#> Note:
#> - Changes are approximate raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by the absolute changes for each variable.

Here is a comparison between the approximation using semfindr::est_change_raw_approx() and semfindr::est_change_raw()

# From semfindr
fit_est_change_raw <- est_change_raw(fit_rerun)
# Plot the differences
library(ggplot2)
tmp1 <- as.vector(t(as.matrix(fit_est_change_raw)))
tmp2 <- as.vector(t(as.matrix(fit_est_change_approx)))
est_change_df <- data.frame(param = rep(colnames(fit_est_change_raw),
                                        nrow(fit_est_change_raw)),
                             est_change = tmp1,
                             est_change_approx = tmp2)
ggplot(est_change_df, aes(x = est_change, y = est_change_approx)) +
  geom_abline(intercept = 0, slope = 1) +
  geom_point(size = 0.8, alpha = 0.5) +
  facet_wrap(~ param) +
  coord_fixed()

The results are pretty similar.

Generalized Cook’s distance (gCD)

We can use the approximate parameter changes to approximate the gCD (see also Tanaka et al., 1991, equation 13, p. 3811):

# Information matrix (Hessian)
information_fit <- lavInspect(fit, what = "information")
# Short cut for computing quadratic form (https://stackoverflow.com/questions/27157127/efficient-way-of-calculating-quadratic-forms-avoid-for-loops)
gcd_approx <- (nobs(fit) - 1) * rowSums(
  (fit_est_change_approx %*% information_fit) * fit_est_change_approx
)

This is implemented in the est_change_approx() function:

fit_est_change_approx <- est_change_approx(fit)
fit_est_change_approx
#> 
#> -- Approximate Standardized Case Influence on Parameter Estimates --
#> 
#>     m1~iv1 m1~iv2  dv~m1 m1~~m1 dv~~dv gcd_approx
#> 16   0.052 -0.038 -0.228 -0.006  0.572      0.372
#> 43  -0.387 -0.249 -0.135  0.201  0.116      0.270
#> 65   0.150  0.189  0.355  0.071  0.148      0.203
#> 85  -0.170  0.211 -0.118  0.315 -0.054      0.187
#> 51   0.405 -0.052  0.094  0.075 -0.046      0.179
#> 34  -0.306 -0.186 -0.110  0.176  0.028      0.163
#> 32  -0.241  0.190 -0.189  0.181 -0.002      0.161
#> 20  -0.234  0.199 -0.140  0.172 -0.034      0.144
#> 18  -0.269  0.035  0.101  0.246 -0.048      0.143
#> 100 -0.001 -0.221 -0.069  0.290 -0.058      0.137
#> 
#> Note:
#> - Changes are approximate standardized raw changes if a case is included.
#> - Only the first 10 case(s) is/are displayed. Set 'first' to NULL to display all cases.
#> - Cases sorted by approximate generalized Cook's distance.

# Compare to exact computation
fit_est_change <- est_change(fit_rerun)
# Plot
gcd_df <- data.frame(
  gcd_exact = fit_est_change[ , "gcd"],
  gcd_approx = fit_est_change_approx[ , "gcd_approx"]
)
ggplot(gcd_df, aes(x = gcd_exact, y = gcd_approx)) +
  geom_abline(intercept = 0, slope = 1) +
  geom_point() +
  coord_fixed()

The approximation tend to underestimate the actual gCD but the rank ordering is almost identical. This is discussed also in Tanaka et al. (1991), who proposed a correction by applying a one-step approximation after the correction (currently not implemented due to the need to recompute scores with updated parameter values).

cor(gcd_df, method = "spearman")
#>            gcd_exact gcd_approx
#> gcd_exact   1.000000   0.999892
#> gcd_approx  0.999892   1.000000

Approximate Change in Fit

The casewise loglikelihood—the contribution to the likelihood function by an observation—can be computed in lavaan, which approximates the change in loglikelihood when an observation is deleted:

lli <- lavInspect(fit, what = "loglik.casewise")
head(lli)
#> [1] -2.776787 -2.034084 -2.154825 -2.248100 -2.793426 -2.049238

Here, \(\ell(\hat \theta)\) will drop 2.78 when observation 1 is deleted. This should approximate \(\ell(\hat \theta_{(-i)})\) as long as \(\hat \theta_{(-i)}\) is not too different from \(\hat \theta\). Here’s a comparison:

# Predicted ll without observation 1
fit@loglik$loglik - lli[1]
#> [1] -289.8272
# Actual ll without observation 1
fit_no1 <- sem(mod, dat[-1, ])
fit_no1@loglik$loglik
#> [1] -289.8156

They are pretty close. To approximate the change in \(\chi^2\), as well as other \(\chi^2\)-based fit indices, we can use the fit_measures_change_approx() function:

chisq_i_approx <- fit_measures_change_approx(fit)
# Compare to the actual chisq when dropping observation 1
c(predict = chisq_i_approx[1, "chisq"] + fitmeasures(fit, "chisq"),
  actual = fitmeasures(fit_no1, "chisq"))
#> predict.chisq  actual.chisq 
#>      6.871042      6.557397

Comparing exact and approximate changes in fit indices

Change in \(\chi^2\)

# Exact measure from semfindr
out <- fit_measures_change(fit_rerun)
# Plot
chisq_change_df <- data.frame(
  chisq_change = out[ , "chisq"],
  chisq_change_approx = chisq_i_approx[ , "chisq"]
)
ggplot(chisq_change_df, aes(x = chisq_change, y = chisq_change_approx)) +
  geom_abline(intercept = 0, slope = 1) +
  geom_point() +
  coord_fixed()

Change in RMSEA

# Plot
rmsea_change_df <- data.frame(
  rmsea_change = out[ , "rmsea"],
  rmsea_change_approx = chisq_i_approx[ , "rmsea"]
)
ggplot(rmsea_change_df, aes(x = rmsea_change, y = rmsea_change_approx)) +
  geom_abline(intercept = 0, slope = 1) +
  geom_point() +
  coord_fixed()

The values aligned reasonably well along the 45-degree line.

Limitations

The approximate approach is tested only for models fitted by maximum likelihood (ML) with normal theory standard errors (the default).
The approximate approach does not yet support multilevel models.

The lavaan object will be checked by approx_check() to see if it is supported. If not, an error will be raised.

References

Cook, R. D., & Weisberg, S. (1982). Residuals and influence in regression. New York: Chapman and Hall. https://conservancy.umn.edu/handle/11299/37076

Tanaka, Y., Watadani, S., & Ho Moon, S. (1991). Influence in covariance structure analysis: With an application to confirmatory factor analysis. Communications in Statistics - Theory and Methods, 20(12), 3805–3821. https://doi.org/10.1080/03610929108830742

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