The Delta Method for Ratio

Applies the Delta method to the ratio of two random variables, \(f(X,Y)=X/Y\), to estimate the expected value, variance, standard error, and confidence interval.

References

• id:sz_dr (2018). Calculating the mean and variance of the ratio of random variables using the Delta method [in Japanese]. If you are human, think more now. https://www.szdrblog.info/entry/2018/11/18/154952

Methods

Public methods

• DeltaMethodForRatio$new()

• DeltaMethodForRatio$get_expected_value()

• DeltaMethodForRatio$get_variance()

• DeltaMethodForRatio$get_squared_standard_error()

• DeltaMethodForRatio$get_standard_error()

• DeltaMethodForRatio$get_confidence_interval()

• DeltaMethodForRatio$get_info()

• DeltaMethodForRatio$compute_expected_value()

• DeltaMethodForRatio$compute_variance()

• DeltaMethodForRatio$compute_confidence_interval()

• DeltaMethodForRatio$clone()

---

Method new()

Initialize a new DeltaMethodForRatio object.

Usage

DeltaMethodForRatio$new(numerator, denominator, bias_correction = FALSE)

Arguments

numerator, denominator

numeric vectors sampled from the distributions of the random variables in the numerator and denominator of the ratio.

bias_correction

logical value indicating whether correction to the mean of the metric is performed using the second-order term of the Taylor expansion. The default is FALSE.

---

Method get_expected_value()

Get the expected value.

Usage

DeltaMethodForRatio$get_expected_value()

Returns

numeric estimate of the expected value of the ratio.

---

Method get_variance()

Get the variance.

Usage

DeltaMethodForRatio$get_variance()

Returns

numeric estimate of the variance of the ratio.

---

Method get_squared_standard_error()

Get the squared standard error.

Usage

DeltaMethodForRatio$get_squared_standard_error()

Returns

numeric estimate of the squared standard error of the ratio.

---

Method get_standard_error()

Get the standard error.

Usage

DeltaMethodForRatio$get_standard_error()

Returns

numeric estimate of the standard error of the ratio.

---

Method get_confidence_interval()

Get the confidence interval.

Usage

DeltaMethodForRatio$get_confidence_interval(
  alternative = c("two.sided", "less", "greater"),
  conf_level = 0.95
)

Arguments

alternative

character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", or "less". You can specify just the initial letter.

conf_level

numeric value specifying the confidence level of the interval. The default is 0.95.

Returns

numeric estimates of the lower and upper bounds of the confidence interval of the ratio.

---

Method get_info()

Get statistical information.

Usage

DeltaMethodForRatio$get_info(
  alternative = c("two.sided", "less", "greater"),
  conf_level = 0.95
)

Arguments

alternative

character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", or "less". You can specify just the initial letter.

conf_level

numeric value specifying the confidence level of the interval. The default is 0.95.

Returns

numeric estimates include the expected value, variance, standard error, and confidence interval.

---

Method compute_expected_value()

Class method to compute the expected value of the ratio using the Delta method.

Usage

DeltaMethodForRatio$compute_expected_value(
  mean1,
  mean2,
  var2,
  cov = 0,
  bias_correction = FALSE
)

Arguments

mean1

numeric value of the mean numerator of the ratio.

mean2

numeric value of the mean denominator of the ratio.

var2

numeric value of the variance of the denominator of the ratio.

cov

numeric value of the covariance between the numerator and denominator of the ratio. The default is 0.

bias_correction

logical value indicating whether correction to the mean of the metric is performed using the second-order term of the Taylor expansion. The default is FALSE.

Returns

numeric estimate of the expected value of the ratio.

---

Method compute_variance()

Class method to compute the variance of the ratio using the Delta method.

Usage

DeltaMethodForRatio$compute_variance(mean1, mean2, var1, var2, cov = 0)

Arguments

mean1

numeric value of the mean numerator of the ratio.

mean2

numeric value of the mean denominator of the ratio.

var1

numeric value of the variance of the numerator of the ratio.

var2

numeric value of the variance of the denominator of the ratio.

cov

numeric value of the covariance between the numerator and denominator of the ratio. The default is 0.

Returns

numeric estimate of the variance of the ratio

---

Method compute_confidence_interval()

Class method to compute the confidence interval of the ratio using the Delta method.

Usage

DeltaMethodForRatio$compute_confidence_interval(
  mean,
  standard_error,
  alternative = c("two.sided", "less", "greater"),
  conf_level = 0.95
)

Arguments

mean

numeric value of the estimated mean of the ratio.

standard_error

numeric value of the estimated standard error of the mean of the ratio.

alternative

character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", or "less". You can specify just the initial letter.

conf_level

numeric value specifying the confidence level of the interval. The default is 0.95.

Returns

numeric estimates of the lower and upper bounds of the confidence interval of the ratio.

---

Method clone()

The objects of this class are cloneable with this method.

Usage

DeltaMethodForRatio$clone(deep = FALSE)

Arguments

deep

Whether to make a deep clone.