arules (version 1.7.15)

confint: Confidence Intervals for Interest Measures for Association Rules

Description

Defines a method to compute confidence intervals for interest measures for association rules.

Usage

# S3 method for rules
confint(
  object,
  parm = "oddsRatio",
  level = 0.95,
  measure = NULL,
  side = c("two.sided", "lower", "upper"),
  method = NULL,
  replications = 1000,
  smoothCounts = 0,
  transactions = NULL,
  ...
)

Value

Returns a matrix with with one row for each rule and the two columns named "LL" and "UL" with the interval boundaries. The matrix has the following additional attributes:

measure

the interest measure.

level

the confidence level

side

the confidence level

smoothCounts

used count smoothing.

method

name of the method to create the interval

desc

description of the used method to calculate the confidence interval. The mentioned references can be found below.

Arguments

object

an object of class rules.

parm, measure

name of the interest measures (see interestMeasure()). measure can be used instead of parm.

level

the confidence level required.

side

Should a two-sided confidence interval or a one-sided limit be returned? Lower returns an interval with only a lower limit and upper returns an interval with only an upper limit.

method

method to construct the confidence interval. The available methods depends on the measure and the most common method is used by default.

replications

number of replications for method "bootstrap". Ignored for other methods.

smoothCounts

pseudo count for addaptive smoothing (Laplace smoothing). Often a pseudo counts of .5 is used for smoothing (see Detail Section).

transactions

transactions used to calculate the contingency-table counts. If supplied, stored rule-quality values are not reused. An independent validation dataset can be supplied to obtain confidence intervals that are not affected by mining and selecting the rules on the same observations.

...

Additional parameters are ignored with a warning.

Fast Confidence Interval Estimation

Fast confidence interval approximations are currently available and used for the measures "support", "count", "confidence", "lift", "oddsRatio", and "phi".

Methods:

• "exact": Exact binomial proportion confidence interval (Clopper & Pearson, 1934).

• "normal": Normal approximation population proportion confidence interval (Wilson, 1927).

• "wilson": Wilson score interval (Wilson, 1927).

• "woolf": Woolf method confidence interval for log of the odds ratio (Woolf, 1955).

• "delta", "log_delta": Delta and Log delta method (Doob, 1935).

• "gart": Haldane-Anscombe-Gart interval. Delta method with count smoothing of .5 (Haldane, 1956).

Available methods by interest measure:

Interest measure	Default fast method	Other available fast methods
"count"	"wilson"	"normal", "exact"
"support"	"wilson"	"normal", "exact"
"confidence"	"delta"	"log_delta", "wilson", "normal", "exact"
"lift"	"delta"	"log_delta"
"oddsRatio"	"woolf"	"gart", "exact"
"phi"	"delta"	None

The "bootstrap" method is also available.

Count Smoothing

All intervals are calculated using count data. Haldan-Anscombe correction (Haldan, 1940; Anscombe, 1956) avoids issues with zero counts by count smoothing (adding .5 to each count). Haldan-Anscombe correction of smoothCounts = 0.5 can be used with any interval method.

The Haldane-Anscombe-Gart interval above (method "gart") applies the delta method with Haldan-Anscombe correction to the odds ratio measure (Haldane, 1956).

Using Validation Data

Confidence intervals calculated from the same transactions used to mine and select rules do not account for the rule-selection process. Their nominal coverage may therefore be too optimistic, especially when many candidate rules are examined.

For confirmatory analysis, rules can be mined using training data and an independent validation transaction set can be supplied using transactions. The contingency-table counts and confidence intervals are then recalculated from the validation data instead of using the quality measures stored with the rules. When many rules are evaluated on the validation data, multiple-comparison adjustments or a further independent test set may still be appropriate.

Author

Michael Hahsler

Details

This method creates a contingency table for each rule and then constructs a confidence interval for the specified measures. Confidence intervals for all interest measures can be assessed using the "bootstrap" method. However, since bootstrapping has to be applied to each rule separately, this can be slow. For some popular measures, faster estimates are available.

References

Wilson, E. B. (1927). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association, 22 (158): 209-212. tools:::Rd_expr_doi("10.1080/01621459.1927.10502953")

Clopper, C.; Pearson, E. S. (1934). "The use of confidence or fiducial limits illustrated in the case of the binomial". Biometrika, 26 (4): 404-413. tools:::Rd_expr_doi("10.1093/biomet/26.4.404")

Doob, J. L. (1935). "The Limiting Distributions of Certain Statistics". Annals of Mathematical Statistics, 6: 160-169. tools:::Rd_expr_doi("10.1214/aoms/1177732594")

Fisher, R.A. (1962). "Confidence limits for a cross-product ratio". Australian Journal of Statistics, 4, 41.

Wilson, E.B. (1927, 6). "Probable inference, the law of succession, and statistical inference". Journal of the American Statistical Association, 22.

Woolf, B. (1955). "On estimating the relation between blood group and diseases". Annals of Human Genetics, 19, 251-253.

Haldane, J.B.S. (1940). "The mean and variance of the moments of chi-squared when used as a test of homogeneity, when expectations are small". Biometrika, 29, 133-134.

Haldane, J.B.S. (1956, 5). "The estimation and significance of the logarithm of a ratio of frequencies". Annals of Human Genetics, 20.

Anscombe, F.J. (1956). "On estimating binomial response relations". Biometrika, 43, 461-464.

See Also

Other interest measures: coverage(), interestMeasure(), is.redundant(), is.significant(), support()

Examples

data("Income")

# mine some rules with the consequent "language in home=english"
rules <- apriori(Income,
  parameter = list(support = 0.5),
  appearance = list(rhs = "language in home=english")
)

# calculate the confidence interval for the rules' odds ratios.
# note that we use Haldane-Anscombe correction (with smoothCounts = .5)
# to avoid issues with 0 counts in the contingency table.
ci <- confint(rules, "oddsRatio", smoothCounts = .5)
ci

# We add the odds ratio (with Haldane-Anscombe correction)
# and the confidence intervals to the quality slot of the rules.
quality(rules) <- cbind(
  quality(rules),
  oddsRatio = interestMeasure(rules, "oddsRatio", smoothCounts = .5),
  oddsRatio = ci
)

rules <- sort(rules, by = "oddsRatio")
inspect(rules)

# use confidence intervals for lift to find rules with a lift significantly larger then 1.
# We set the confidence level to 95%, create a one-sided interval and check
# if the interval does not cover 1 (i.e., the lower limit is larger than 1).
ci <- confint(rules, "lift", level = 0.95, side = "lower")
ci

inspect(rules[ci[, "LL"] > 1])

# For confirmatory analysis, mine rules on training data and calculate
# confidence intervals using independent validation data.
set.seed(1234)
training_ids <- sample(seq_along(Income), floor(.7 * length(Income)))
training <- Income[training_ids]
validation <- Income[-training_ids]

validation_rules <- apriori(training,
  parameter = list(support = .5),
  appearance = list(rhs = "language in home=english")
)
validation_ci <- confint(validation_rules,
  "lift",
  transactions = validation,
  side = "lower"
)
inspect(validation_rules[validation_ci[, "LL"] > 1])