Optimal Subsampling Methods for Generalized Linear Models

Draw subsample from full dataset and fit a generalized linear model (GLM) on the subsample. For a quick start, refer to the vignette.

Usage

ssp.glm(
  formula,
  data,
  subset = NULL,
  n.plt,
  n.ssp,
  family = "binomial",
  criterion = "optL",
  sampling.method = "poisson",
  likelihood = "weighted",
  control = list(...),
  contrasts = NULL,
  ...
)

Arguments

formula

A model formula object of class "formula" that describes the model to be fitted.

data

A data frame containing the variables in the model. Denote $N$ as the number of observations in data.

subset

An optional vector specifying a subset of observations from data to use for the analysis. This subset will be viewed as the full data.

n.plt

The pilot subsample size (first-step subsample size). This subsample is used to compute the pilot estimator and estimate the optimal subsampling probabilities.

n.ssp

The expected size of the optimal subsample (second-step subsample). For sampling.method = 'withReplacement', The exact subsample size is n.ssp. For sampling.method = 'poisson', n.ssp is the expected subsample size.

family

family can be a character string naming a family function, a family function or the result of a call to a family function.

criterion

The choices include optA, optL(default), LCC and uniform.

optA Minimizes the trace of the asymptotic covariance matrix of the subsample estimator.
optL Minimizes the trace of a transformation of the asymptotic covariance matrix. The computational complexity of optA is $O(N d^2)$ while that of optL is $O(N d)$.
LCC Local Case-Control sampling probability, used as a baseline subsampling strategy.
uniform Assigns equal subsampling probability $\frac{1}{N}$ to each observation, serving as a baseline subsampling strategy.

sampling.method

The sampling method to use. Options include withReplacement and poisson (default). withReplacement draws exactly n.ssp subsamples from size $N$ full dataset with replacement, using the specified subsampling probabilities. poisson draws observations independently by comparing each subsampling probability with a realization of uniform random variable $U(0,1)$.

Differences between methods:

Sample size: withReplacement draws exactly n.ssp subsamples while poisson draws subsamples with expected size n.ssp, meaning the actual size may vary.
Memory usage: withReplacement requires the entire dataset to be loaded at once, while poisson allows for processing observations sequentially (will be implemented in future version).
Estimator performance: Theoretical results show that the poisson tends to get a subsample estimator with lower asymptotic variance compared to the withReplacement

likelihood

The likelihood function to use. Options include weighted (default) and logOddsCorrection. A bias-correction likelihood function is required for subsample since unequal subsampling probabilities introduce bias.

weighted Applies a weighted likelihood function where each observation is weighted by the inverse of its subsampling probability.
logOddsCorrection This lieklihood is available only for logistic regression model (i.e., when family is binomial or quasibinomial). It uses a conditional likelihood, where each element of the likelihood represents the probability of $Y=1$, given that this subsample was drawn.

control

The argument control contains two tuning parameters alpha and b.

alpha $\in [0,1]$ is the mixture weight of the user-assigned subsampling probability and uniform subsampling probability. The actual subsample probability is $\pi = (1-\alpha)\pi^{opt} + \alpha \pi^{uni}$. This protects the estimator from extreme small subsampling probability. The default value is 0.
b is a positive number which is used to constaint the poisson subsampling probability. b close to 0 results in subsampling probabilities closer to uniform probability $\frac{1}{N}$. b=2 is the default value. See relevant references for further details.

contrasts

An optional list. It specifies how categorical variables are represented in the design matrix. For example, contrasts = list(v1 = 'contr.treatment', v2 = 'contr.sum').

...

A list of parameters which will be passed to svyglm().

Value

ssp.glm returns an object of class "ssp.glm" containing the following components (some are optional):

model.call: The original function call.
coef.plt: The pilot estimator. See Details for more information.
coef.ssp: The estimator obtained from the optimal subsample.
coef: The weighted linear combination of coef.plt and coef.ssp. The combination weights depend on the relative size of n.plt and n.ssp and the estimated covariance matrices of coef.plt and coef.ssp. We blend the pilot subsample information into optimal subsample estimator since the pilot subsample has already been drawn. The coefficients and standard errors reported by summary are coef and the square root of diag(cov).
cov.ssp: The covariance matrix of coef.ssp.
cov: The covariance matrix of coef.
index.plt: Row indices of pilot subsample in the full dataset.
index.ssp: Row indices of of optimal subsample in the full dataset.
N: The number of observations in the full dataset.
subsample.size.expect: The expected subsample size, equals to n.ssp for ssp.glm. Note that for other functions, such as ssp.relogit, this value may differ.
terms: The terms object for the fitted model.

Details

A pilot estimator for the unknown parameter $\beta$ is required because both optA and optL subsampling probabilities depend on $\beta$. There is no "free lunch" when determining optimal subsampling probabilities. Fortunately the pilot estimator only needs to satisfy mild conditions. For logistic regression, this is achieved by drawing a size n.plt subsample with replacement from full dataset. The case-control subsample probability is applied, that is, $\pi_i = \frac{1}{2N_1}$ for $Y_i=1$ and $\pi_i = \frac{1}{2N_0}$ for $Y_i=0$, $i=1,...,N$, where$N_0$ and $N_1$ are the counts of observations with $Y = 0$ and $Y = 1$, respectively. For other families, uniform subsampling probabilities are applied. Typically, n.plt is relatively small compared to n.ssp.

When criterion = 'uniform', there is no need to compute the pilot estimator. In this case, a size n.plt + n.ssp subsample will be drawn with uniform sampling probability and coef is the corresponding estimator.

As suggested by survey::svyglm(), for binomial and poisson families, use family=quasibinomial() and family=quasipoisson() to avoid a warning "In eval(family$initialize) : non-integer #successes in a binomial glm!". The quasi versions of the family objects give the same point estimates and suppress the warning. Since subsampling methods only rely on point estimates from svyglm() for further computation, using the quasi families does not introduce any issues.

For Gamma family, ssp.glm returns only the estimation of coefficients, as the dispersion parameter is not estimated.

References

Wang, H. (2019). More efficient estimation for logistic regression with optimal subsamples. Journal of machine learning research, 20(132), 1-59.

Ai, M., Yu, J., Zhang, H., & Wang, H. (2021). Optimal subsampling algorithms for big data regressions. Statistica Sinica, 31(2), 749-772.

Wang, H., & Kim, J. K. (2022). Maximum sampled conditional likelihood for informative subsampling. Journal of machine learning research, 23(332), 1-50.

Examples

# logistic regression
set.seed(2)
N <- 1e4
beta0 <- rep(-0.5, 7)
d <- length(beta0) - 1
corr <- 0.5
sigmax  <- matrix(corr, d, d) + diag(1-corr, d)
X <- MASS::mvrnorm(N, rep(0, d), sigmax)
Y <- rbinom(N, 1, 1 - 1 / (1 + exp(beta0[1] + X %*% beta0[-1])))
data <- as.data.frame(cbind(Y, X))
formula <- Y ~ .
n.plt <- 500
n.ssp <- 1000
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial',
criterion = "optL",
sampling.method = 'poisson',
likelihood = "logOddsCorrection")
summary(subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "quasibinomial", criterion = "optL", sampling.method = "poisson", 
#>     likelihood = "logOddsCorrection")
#> 
#> Subsample Size:
#>                                 
#> 1       Total Sample Size  10000
#> 2 Expected Subsample Size   1000
#> 3   Actual Subsample Size   1034
#> 4   Unique Subsample Size   1034
#> 5  Expected Subample Rate    10%
#> 6    Actual Subample Rate 10.34%
#> 7    Unique Subample Rate 10.34%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept  -0.5430     0.0603 -9.0098  <0.0001
#> V2         -0.5562     0.0743 -7.4864  <0.0001
#> V3         -0.4597     0.0726 -6.3341  <0.0001
#> V4         -0.4561     0.0761 -5.9945  <0.0001
#> V5         -0.5714     0.0755 -7.5644  <0.0001
#> V6         -0.4958     0.0721 -6.8725  <0.0001
#> V7         -0.4962     0.0736 -6.7385  <0.0001
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial', 
criterion = "optL",
sampling.method = 'withReplacement', 
likelihood = "weighted")
summary(subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "quasibinomial", criterion = "optL", sampling.method = "withReplacement", 
#>     likelihood = "weighted")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size  1000
#> 3   Actual Subsample Size  1000
#> 4   Unique Subsample Size   901
#> 5  Expected Subample Rate   10%
#> 6    Actual Subample Rate   10%
#> 7    Unique Subample Rate 9.01%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept  -0.5201     0.0649 -8.0143  <0.0001
#> V2         -0.6526     0.0853 -7.6500  <0.0001
#> V3         -0.4247     0.0864 -4.9176  <0.0001
#> V4         -0.2804     0.0841 -3.3329   0.0009
#> V5         -0.5186     0.0792 -6.5508  <0.0001
#> V6         -0.6281     0.0837 -7.5026  <0.0001
#> V7         -0.5878     0.0836 -7.0348  <0.0001
Uni.subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'quasibinomial', 
criterion = 'uniform')
summary(Uni.subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "quasibinomial", criterion = "uniform")
#> 
#> Subsample Size:
#>                                 
#> 1       Total Sample Size  10000
#> 2 Expected Subsample Size   1500
#> 3   Actual Subsample Size   1462
#> 4   Unique Subsample Size   1462
#> 5  Expected Subample Rate    15%
#> 6    Actual Subample Rate 14.62%
#> 7    Unique Subample Rate 14.62%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept  -0.5165     0.0720 -7.1700  <0.0001
#> V2         -0.7625     0.1000 -7.6253  <0.0001
#> V3         -0.3873     0.0919 -4.2152  <0.0001
#> V4         -0.3826     0.0976 -3.9188  <0.0001
#> V5         -0.5533     0.0914 -6.0567  <0.0001
#> V6         -0.3779     0.0991 -3.8143   0.0001
#> V7         -0.5427     0.0923 -5.8806  <0.0001
####################
# poisson regression
set.seed(1)
N <-  1e4
beta0 <- rep(0.5, 7)
d <- length(beta0) - 1
X <- matrix(runif(N * d), N, d)
epsilon <- runif(N)
lambda <- exp(beta0[1] + X %*% beta0[-1])
Y <- rpois(N, lambda)
data <- as.data.frame(cbind(Y, X))
formula <- Y ~ .
n.plt <- 200
n.ssp <- 600
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson',
criterion = "optL", 
sampling.method = 'poisson',
likelihood = "weighted")
summary(subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "poisson", criterion = "optL", sampling.method = "poisson", 
#>     likelihood = "weighted")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size   600
#> 3   Actual Subsample Size   681
#> 4   Unique Subsample Size   681
#> 5  Expected Subample Rate    6%
#> 6    Actual Subample Rate 6.81%
#> 7    Unique Subample Rate 6.81%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept   0.5662     0.0568  9.9711  <0.0001
#> V2          0.4893     0.0357 13.7219  <0.0001
#> V3          0.5126     0.0370 13.8583  <0.0001
#> V4          0.4253     0.0351 12.1181  <0.0001
#> V5          0.4972     0.0397 12.5128  <0.0001
#> V6          0.4930     0.0383 12.8792  <0.0001
#> V7          0.4871     0.0377 12.9209  <0.0001
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson', 
criterion = "optL", 
sampling.method = 'withReplacement',
likelihood = "weighted")
summary(subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "poisson", criterion = "optL", sampling.method = "withReplacement", 
#>     likelihood = "weighted")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size   600
#> 3   Actual Subsample Size   600
#> 4   Unique Subsample Size   568
#> 5  Expected Subample Rate    6%
#> 6    Actual Subample Rate    6%
#> 7    Unique Subample Rate 5.68%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept   0.5543     0.0590  9.3997  <0.0001
#> V2          0.4505     0.0402 11.2074  <0.0001
#> V3          0.4643     0.0423 10.9781  <0.0001
#> V4          0.5392     0.0420 12.8471  <0.0001
#> V5          0.4497     0.0425 10.5933  <0.0001
#> V6          0.4529     0.0415 10.9180  <0.0001
#> V7          0.5495     0.0395 13.9048  <0.0001
Uni.subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'poisson', 
criterion = 'uniform')
summary(Uni.subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "poisson", criterion = "uniform")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size   800
#> 3   Actual Subsample Size   820
#> 4   Unique Subsample Size   820
#> 5  Expected Subample Rate    8%
#> 6    Actual Subample Rate  8.2%
#> 7    Unique Subample Rate  8.2%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept   0.4226     0.0591  7.1479  <0.0001
#> V2          0.5789     0.0431 13.4381  <0.0001
#> V3          0.4852     0.0425 11.4087  <0.0001
#> V4          0.5401     0.0427 12.6439  <0.0001
#> V5          0.4419     0.0432 10.2351  <0.0001
#> V6          0.5156     0.0436 11.8327  <0.0001
#> V7          0.5311     0.0418 12.6998  <0.0001
##################
# gamma regression
set.seed(1)
N <- 1e4
p <- 3
beta0 <- rep(0.5, p + 1)
d <- length(beta0) - 1
shape <- 2
X <- matrix(runif(N * d), N, d)
link_function <- function(X, beta0) 1 / (beta0[1] + X %*% beta0[-1])
scale <- link_function(X, beta0) / shape
Y <- rgamma(N, shape = shape, scale = scale)
data <- as.data.frame(cbind(Y, X))
formula <- Y ~ .
n.plt <- 200
n.ssp <- 1000
subsampling.results <- ssp.glm(formula = formula, 
data = data, 
n.plt = n.plt,
n.ssp = n.ssp,
family = 'Gamma',
criterion = "optL", 
sampling.method = 'poisson',
likelihood = "weighted")
summary(subsampling.results)
#> Model Summary
#> 
#> Call:
#> 
#> ssp.glm(formula = formula, data = data, n.plt = n.plt, n.ssp = n.ssp, 
#>     family = "Gamma", criterion = "optL", sampling.method = "poisson", 
#>     likelihood = "weighted")
#> 
#> Subsample Size:
#>                                
#> 1       Total Sample Size 10000
#> 2 Expected Subsample Size  1000
#> 3   Actual Subsample Size   989
#> 4   Unique Subsample Size   989
#> 5  Expected Subample Rate   10%
#> 6    Actual Subample Rate 9.89%
#> 7    Unique Subample Rate 9.89%
#> 
#> Coefficients:
#> 
#>           Estimate Std. Error z value Pr(>|z|)
#> Intercept   0.5099     0.0510  9.9893  <0.0001
#> V2          0.4732     0.0603  7.8469  <0.0001
#> V3          0.4040     0.0621  6.5035  <0.0001
#> V4          0.4945     0.0681  7.2599  <0.0001