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The '''ratio estimator''' is a [[statistical parameter]] and is defined to be the [[ratio]] of [[Arithmetic mean|mean]]s of two [[variate]]s. Ratio estimates are [[Bias (statistics)|bias]]ed and corrections must be made when they are used in experimental or survey work. The ratio estimates are asymmetrical and symmetrical tests such as the [[t test]] should not be used to generate confidence intervals. 

It is also worth noting that the bias is of the order ''O''(1/''n'') (see [[big O notation]]) so as the sample size (''n'') increases, the bias will asymptotically approach 0. Therefore, the estimator is approximately unbiased for large sample sizes.

==Definition==

Assume there are two characteristics – ''x'' and ''y'' – that can be observed for each sampled element in the data set. The ratio ''R'' is

: <math> R = \bar{\mu}_y  / \bar{\mu}_x \, </math>

The ratio estimate of a value of the ''y'' variate (''θ''<sub>''y''</sub>) is

: <math> \theta_y = R \theta_x \, </math>

where ''θ''<sub>''x''</sub> is the corresponding value of the ''x'' variate. ''θ''<sub>''y''</sub> is known to be asymptotically normally distributed.<ref name=Scott1981>Scott AJ, Wu CFJ (1981) On the asymptotic distribution of ratio and regression
estimators. JASA 76: 98–102</ref>

==Statistical properties==
{{seealso|ratio distribution}}
The sample ratio (''r'') is estimated from the sample

: <math> r = \frac{ \bar{ y } }{ \bar{ x } } = \frac{ \sum_{ i = 1 }^n y }{ \sum_{ i = 1 }^n x }</math>

That the ratio is biased can be shown with [[Jensen's inequality]] as follows:

: <math> E\left( \frac{ x }{ y } \right) = E\left( x \frac{ 1 }{ y } \right) = E( x )E\left( \frac{ 1 }{ y } \right) \ge E(x)\frac{ 1 }{ E( y ) } = \frac{ E( x )}{ E( y ) } </math>

Under simple random sampling the bias is of the order ''O''( ''n''<sup>−1</sup> ). An upper bound on the relative bias of the estimate is provided by the [[coefficient of variation]] (the ratio of the [[standard deviation]] to the [[mean]]).<ref name=Cochran1977>Cochran WG (1977) Sampling techniques. New York: John Wiley & Sons</ref> Under simple random sampling the relative bias is ''O''( ''n''<sup>-1/2</sup> ).

===Correction of the mean's bias===

The correction methods, depending on the distributions of the ''x'' and ''y'' variates, differ in their efficiency making it difficult to recommend an overall best method. Because the estimates of ''r'' are biased a corrected version should be used in all subsequent calculations.

A correction of the bias accurate to the first order is<ref name=Hartley1954>Hartley HO, Ross A (1954) Unbiased ratio estimators. Nature 174: 270–271</ref> 

: <math> r_\mathrm{ corr } = r - \frac{ s_{ [ y / x ] x } }{ m_x } </math>

where ''m''<sub>''x''</sub> is the mean of the variate ''x'' and ''s''<sub>''ab''</sub> is the [[covariance]] between ''a'' and ''b''.

To simplify the notation ''s''<sub>ab</sub> will be used subsequently to denote the covariance between the variates ''a'' and ''b''.

Another estimator based on the [[Taylor expansion]] is

: <math> r_\mathrm{ corr } = r + \frac{ 1 }{ n }( 1 - \frac{ n - 1 }{ N - 1 } ) \frac{ r s_x^2 - \rho s_x s_y }{ m_x^2 } </math>

where ''n'' is the sample size, ''N'' is the population size, ''m''<sub>''x''</sub> is the mean of the variate ''x'', ''s''<sub>''x''</sub><sup>2</sup> and ''s''<sub>''y''</sub><sup>2</sup> are the sample [[variance]]s of the ''x'' and ''y'' variates respectively and ''ρ'' is the sample correlation between the ''x'' and ''y'' variates.

A computationally simpler but slightly less accurate version of this estimator is

: <math> r_\mathrm{ corr } = r - \frac{ N - n }{ N } \frac{ ( r s_x^2 - \rho s_x s_y ) }{ n m_x^2 } </math>

where ''N'' is the population size, ''n'' is the sample size, ''m''<sub>''x''</sub> is the mean of the ''x'' variate, ''s''<sub>''x''</sub><sup>2</sup> and ''s''<sub>''y''</sub><sup>2</sup> are the sample [[variance]]s of the ''x'' and ''y'' variates respectively and ''ρ'' is the sample correlation between the ''x'' and ''y'' variates. These versions differ only in the factor in the denominator ( ''N'' - 1 ). For a large ''N'' the difference is negligible.  

A second-order correction is<ref name=Ogliore2011>Ogliore RC, Huss GR, Nagashima K (2011) Ratio estimation in SIMS analysis. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 269 (17) 1910–1918</ref>

: <math> r_\mathrm{ corr } = r \left[ 1 + \frac{ 1 }{ n } \left( \frac{ 1 }{ m_x } - \frac{ s_{ xy } }{ m_x m_y } \right) + \frac{ 1 }{ n^2 } \left( \frac{ 2 }{ m_x^2 } - \frac{ s_{ xy } }{ m_x m_y } \left[ 2 + \frac{ 3 }{ m_x } \right] + \frac{ s_{ x^2 y } }{ m_x^2 m_y } \right) \right]</math>

Other methods of bias correction have also been proposed. To simplify the notation the following variables will be used

: <math> \theta = \frac{ 1 }{ n } - \frac{ 1 }{ N } </math> 
: <math> c_x^2 = \frac{ s_x^2 }{ m_x^2 } </math> 
: <math> c_{ xy } = \frac{ s_{xy} }{ m_x m_y } </math> 

Pascual's estimator:<ref name=Pascual1961>Pascual JN (1961) Unbiased ratio estimators in stratified sampling. JASA 56(293):70–87</ref>

: <math> r_\mathrm{ corr } = r + \frac{ N - 1 }{ N } \frac{ m_y - r m_x }{ n - 1 }</math>

Beale's estimator:<ref name=Beale1962>Beale EML (1962) Some use of computers in operational research. Industrielle Organization 31: 27-28</ref>

: <math> r_\mathrm{ corr } = r \frac{ 1 + \theta c_{ xy } }{ 1 + \theta c_x^2 }</math>

Tin's estimator:<ref name=Tin1965>Tin M (1965) Comparison of some ratio estimators. JASA 60: 294–307</ref>

: <math> r_\mathrm{ corr } = r \left( 1 + \theta \left(  c_{ xy } - c_x^2 \right) \right)</math>

Sahoo's estimator:<ref name=Sahoo1983>Sahoo LN (1983). On a method of bias reduction in ratio estimation. J Statist Res 17:1—6</ref>

: <math> r_\mathrm{ corr } = \frac{ r }{ 1 + \theta ( c_x^2 - c_{ xy } ) } </math>

Sahoo has also proposed a number of additional estimators:<ref name=Sahoo1987>Sahoo LN (1987) On a class of almost unbiased estimators for population ratio. Statistics 18: 119-121</ref>

: <math> r_\mathrm{ corr } = r ( 1 + \theta c_{ xy } ) ( 1 - \theta c_x^2 ) </math>

: <math> r_\mathrm{ corr } = \frac{ r ( 1 - \theta c_x^2 ) }{ 1 - \theta c_{ xy } } </math>

: <math> r_\mathrm{ corr } = \frac{ r }{ ( 1 + \theta c_{ xy } )( 1 + \theta c_x^2 ) } </math>

If ''m''<sub>''x''</sub> and ''m''<sub>''y''</sub> are both greater than 10, then the following approximation is correct to order O( ''n''<sup>−3</sup> ).<ref name=Ogliore2011>Ogliore RC, Huss GR, Nagashima K (2011) Ratio estimation in SIMS analysis. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 269 (17) 1910–1918</ref>

: <math> r_\mathrm{corr} = r \left[ 1 - \frac{ 2 }{ n^2 m_x } \left( \frac{ 1 }{ m_x } - \frac{ s_{ xy } }{ m_x m_y } \right) \left( 1 + \frac{ 13 }{ 2n } + \frac{ 8 }{ n m_x } \right) \right]</math>

An asymptotically correct estimator is<ref name=vanKempen2000>van Kempen GMP, van Vliet LJ (2000) Mean and variance of ratio estimators used in fluorescence ratio imaging. Cytometry 39:300–305</ref>

: <math> r_\mathrm{ corr } = r + c_x^2 \frac{ m_y }{ m_x } - \frac{ s_{ xy } }{ m_x^2 } </math>

==Jackknife estimation==

A jackknife estimate of the ratio is less biased that the naive form. A jackknife estimator of the ratio is 

: <math> r_\mathrm{corr} = nr - \frac{ n - 1 }{ n } \sum_{ i \ne j 1 }^n r_i </math>

where ''n'' is the size of the sample and the ''r''<sub>i</sub> are estimated with the omission of one pair of variates at a time.<ref name=Choquet1999>Choquet D, L'ecuyer P, Léger C (1999) Bootstrap confidence intervals for ratios of expectations. ACM Transactions on Modeling and Computer Simulation - TOMACS 9 (4) 326-348 DOI: 10.1145/352222.352224</ref>

An alternative method is to divide the sample into ''g'' groups each of size ''p'' with ''n'' = ''pg''.<ref name=Durbin1959>Durbin J (1959) A note on the application of Quenouille's method of bias reduction to estimation of ratios. Biometrika 46: 477-480</ref> Let ''r''<sub>i</sub> be the estimate of the ''i''<sup>th</sup> group. Then the estimator 

: <math> r_\mathrm{corr} = gr - \frac{ g - 1 }{ g } \sum_{ i = 1 }^g r_i </math>

has a bias of at most ''O''( ''n''<sup>−2</sup> ). 

Other estimators based on the division of the sample into ''g'' groups are:<ref name=Mickey1959>Mickey MR (1959) Some finite population unbiased ratio and regression estimators. JASA 54: 596–612</ref> 

: <math> r_\mathrm{ corr } = \frac{ g }{ g + 1 } r - \frac{ 1 }{ g ( g - 1 ) } \sum_{ i = 1 }^g r_i </math>

: <math> r_\mathrm{ corr } = \bar{ r } +\frac{ n }{ n - 1 } \frac{ m_y - \bar{ r } m_x }{ m_x } </math>

: <math> r_\mathrm{ corr } = \bar{ r_g } + \frac{ g ( m_y - \bar{ r_g } m_x ) }{ m_x } </math>

where <math> \bar{ r } </math> is the mean of the ratios ''r''<sub>''g''</sub> of the ''g'' groups and  

: <math> \bar{ r_g } = \sum \frac{ r_i^{'} }{ g }</math>

where ''r''<sub>''i''</sub><sup>'</sup> is the value of the sample ratio with the ''i''<sup>th</sup> group omitted.

===Other methods of estimation===

Other methods of estimating a ratio estimator include [[maximum likelihood]] and [[bootstrapping (statistics)|bootstrapping]].<ref name=Choquet1999>Choquet D, l'Ecuyer P, Leger C(1999) Bootstrap confidence intervals for ratios of expectations.</ref>

==Estimate of total==

The estimated total of the ''y'' variate ( ''τ''<sub>''y''</sub> ) is 

: <math> \tau_y = r \tau_x </math>

where ( ''τ''<sub>''x''</sub> ) is the total of the ''x'' variate.

==Variance estimates==

The variance of the sample ratio is approximately:

: <math> \operatorname{ var }( r ) = \frac{ 1 }{ s_x^2 + m_x^2 } \left[ ( s_y^2 - s_{ x^2 [ y^2 / x^2 ] } ) - ( s_{ x [ y / x ] } )^2 +2 m_y s_{ x[ y / x ] } - \frac{ s_x^2 }{ m_x^2 }( m_y - s_{ x[ y / x ] }^2) \right] </math>

where ''s''<sub>''x''</sub><sup>2</sup> and ''s''<sub>''y''</sub><sup>2</sup> are the variances of the ''x'' and ''y'' variates respectively, ''m''<sub>''x''</sub> and ''m''<sub>''y''</sub> are the means of the ''x'' and ''y'' variates respectively and ''s''<sub>''ab''</sub> is the covariance of ''a'' and ''b''.

Although the approximate variance estimator of the ratio given below is biased, if the sample size is large, the bias in this estimator is negligible. 

: <math> \operatorname{ var }( r ) = \frac{ N - n }{ N } \frac{ 1 }{ m_x^2 } \frac{ \sum_{ i = 1 }^n( y_i - rx_i ) }{ n - 1 } </math>

where ''N'' is the population size, ''n'' is the sample size and ''m''<sub>''x''</sub> is the mean of the ''x'' variate.

Another estimator of the variance based on the [[Taylor expansion]] is

: <math> \operatorname{ var }( r ) = \frac{ 1 }{ n } ( 1 - \frac{ n - 1 }{ N - 1 } ) \frac{ r^2 s_x^2 + s_y^2 - 2 r \rho s_x s_y }{ m_x^2 } ) </math>

where ''n'' is the sample size, ''N'' is the population size and ''ρ'' is the correlation coefficient between the ''x'' and ''y'' variates.

An estimate accurate to O( ''n''<sup>−2</sup> ) is<ref name=vanKempen2000>van Kempen GMP, van Vliet LJ (2000) Mean and variance of ratio estimators used in fluorescence ratio imaging. Cytometry 39:300–305</ref>

: <math> \operatorname{ var }( r ) = \frac{ 1 }{ n }\left[ \frac{ s_y^2 }{ m_x^2 } + \frac{ m_y^2 s_x^2  }{ m_x^4 } - \frac{ 2m_y s_{ xy } }{ m_x^3 } \right] </math>

An estimator accurate to O( ''n''<sup>−3</sup> ) is<ref name=Ogliore2011>Ogliore RC, Huss GR, Nagashima K (2011) Ratio estimation in SIMS analysis. Nuclear Instruments and Methods in Physics Research Section B: Beam Interactions with Materials and Atoms 269 (17) 1910–1918</ref>

: <math> \operatorname{ var }( r ) = r^2 \left[ \frac{ 1 }{ n } \left( \frac{ 1 }{ m_x } + \frac{ 1 }{ m_y } - \frac{ 2 s_{ xy } }{ m_x m_y } \right) + \frac{ 1 }{ n^2 } \left( \frac{ 6 }{ m_x^2 } + \frac{ 3 }{ m_x m_y } + s_{ xy }\left[ \frac{ 4 }{ m_y^2 } - \frac{ 8 }{ m_x m_y } - \frac{ 16 }{ m_x^2 m_y } + \frac{ 5 s_{ xy } }{ m_x^2 m_y^2 } \right] + \frac{ 4 s_{ x^2 y } }{ m_x^2 m_y } - \frac{ 2 s_{ x y^2 }}{ m_x m_y^2 } \right) \right]</math>

A jackknife estimator of the variance is

: <math> \operatorname{ var }( r ) = \frac{ 1 }{ n ( n - 1 ) } \sum_{ i \ne j }^n ( r_i - r_J ) </math>

where ''r''<sub>i</sub> is the ratio with the ''i''<sup>th</sup> pair of variates omitted and ''r''<sub>J</sub> is the jackknife estimate of the ratio.<ref name=Choquet1999>Choquet D, L'ecuyer P, Léger C (1999) Bootstrap confidence intervals for ratios of expectations. ACM Transactions on Modeling and Computer Simulation - TOMACS 9 (4) 326-348 DOI: 10.1145/352222.352224</ref>  

;Variance of total

The variance of the estimated total is

:<math> \operatorname{ var }( \tau_y ) = \tau_y^2 \operatorname{ var }( r ) </math>

;Variance of mean

The variance of the estimated mean of the ''y'' variate is 

: <math> \operatorname{ var }( \bar{ y } ) = m_x^2 \operatorname{ var }( r ) = \frac{ N - n }{ N } \frac{ 1 }{ m_x^2 } \frac{ \sum_{ i = 1 }^n( y_i - rx_i ) }{ n - 1 } 

= \frac{ N - n }{ N } \frac{ ( s_y^2 +r^2 s_x^2 - 2r \rho s_x s_y ) }{ n } </math>

where ''m''<sub>''x''</sub> is the mean of the ''x'' variate, ''s''<sub>''x''</sub><sup>2</sup> and ''s''<sub>''y''</sub><sup>2</sup> are the sample variances of the ''x'' and ''y'' variates respectively and ''ρ'' is the sample correlation between the ''x'' and ''y'' variates.

==Skewness==

The [[skewness]] and the [[kurtosis]] of the ratio depend on the distributions of the ''x'' and ''y'' variates. Estimates have been made of these parameters for [[normal distribution|normally distributed]] ''x'' and ''y'' variates but for other distributions no expressions have yet been derived. It has been found that in general ratio variables are skewed to the right, are [[leptokurtic]] and their nonnormality is increased when magnitude of the denominator's [[coefficient of variation]] is increased.

For normally distributed ''x'' and ''y'' variates the skewness of the ratio is approximately<ref name=Tin1965>Tin M (1965) Comparison of some ratio estimators. JASA 60 (309) 294–307</ref>

: <math> \gamma = \left( \frac{ m_y \omega }{ \sqrt{ n m_x m_y \omega^2 + m_x^2 m_y } } \right)\left(  6 + \frac{ 1 }{ n m_x } \left[ 44 + \frac{ 1 }{ 1 + \omega^2 m_y / m_x } \right] \right)</math>

where 

: <math> \omega = 1 - m_x \operatorname{cov}( x, y ) \, </math>

===Effect on confidence intervals===

Because the ratio estimate is generally skewed confidence intervals created with the variance and symmetrical tests such as the t test are incorrect.<ref name=Choquet1999>Choquet D, l'Ecuyer P, Leger C(1999) Bootstrap confidence intervals for ratios of expectations.</ref> These confidence intervals tend to overestimate the size of the left confidence interval and underestimate the size of the right.

If the ratio estimator is [[unimodal]] (which is frequently the case) then a conservative estimate of the 95% confidence intervals can be made with the [[Vysochanskiï–Petunin inequality]].

==Alternative methods of bias reduction==

An alternative method of reducing or eliminating the bias in the ratio estimator is to alter the method of sampling. The variance of the ratio using these methods differs from the estimates given previously.

===Lahiri's method===

Lahiri introduced the first of these sampling schemes in 1951.<ref name=Lahiri1951>Lahiri DB (1951) A method of sample selection providing unbiased ratio estimates. Bull Int Stat Inst 33: 133–140</ref> 

Choose a number ''M'' ≥ max( ''x''<sub>1</sub>, ..., ''x''<sub>N</sub>) where ''N'' is the population size. Chose one of these elements (''x''<sub>i</sub>). Chose ''u'' at random from a [[uniform distribution (continuous)|uniform distribution]] ''U''(0, 1). If ''uM'' ≤ ''x''<sub>i</sub>, then ''x''<sub>i</sub> is retained in the sample. If not then it is rejected and a new element is chosen. Repeat this process ''N'' times. The same process is carried out with the ''y'' variate. Then the ratio of the sum of the ''y'' variates and the sum of the ''x'' variates chosen in this fashion is an unbiased estimate of the ratio estimator. 

In symbols we have

: <math> r = \frac { \sum y_i }{ \sum x_i } </math>

where ''x''<sub>i</sub> and ''y''<sub>i</sub> are chosen according to the scheme described above.

===Midzuno-Sen's method===

In 1952 Midzuno and Sen independently described a sampling scheme that provides an unbiased estimator of the ratio.<ref name=Midzuno1952>Midzuno H (1952) On the sampling system with probability proportional to the sum of the sizes. Ann Inst Stat Math 3: 99-107</ref><ref name=Sen1952>Sen AR (1952) Present status of probability sampling and its use in the estimation of a characteristic. Econometrika 20-103</ref>

The first sample is chosen with probability proportional to the size of the ''x'' variate. The remaining ''n'' - 1 samples are chosen at random without replacement from the remaining ''N'' - 1 members in the population. The probability of selection under this scheme is

: <math> P = \frac{ \sum x_i } { { N - 1 \choose n - 1 } X } </math>

where ''X'' is the sum of the ''N'' ''x'' variates and the ''x''<sub>i</sub> are the ''n'' members of the sample.

The ratio estimator given by this scheme is unbiased.

==Ordinary least squares regression==

If a linear relationship between the ''x'' and ''y'' variates exists and the [[Linear regression|regression]] equation passes through the origin then the estimated variance of the regression equation is always less than that of the ratio estimator. The precise relationship between the variances depends on the linearity of the relationship between the ''x'' and ''y'' variates: when the relationship is other than linear the ratio estimate may have a lower variance than that estimated by regression.

==Uses==

Although the ratio estimator may be of use in a number of settings it is of particular use in two cases:

* when the variates ''x'' and ''y'' are highly [[correlate]]d through the [[Origin (mathematics)|origin]]

* when the total population size is unknown

==History==

The first known use of the ratio estimator was by [[John Graunt]] in [[England]] who in 1662 was the first to estimate the ratio ''y''/''x'' where ''y'' represented the total population and ''x'' the known total number of registered births in the same areas during the preceding year.

Later Messance (~1765) and Moheau (1778) published very carefully prepared estimates for [[France]] based on enumeration of population in certain districts and on the count of births, deaths and marriages as reported for the whole country. The districts from which the ratio of inhabitants to birth was determined only constituted a sample. 

In 1802 [[Laplace]] wished to estimate the population of France. No [[population census]] had been carried out and Laplace lacked the resources to count every individual. Instead he sampled 30 [[parish]]es whose total number of inhabitants was 2,037,615. The parish baptismal registrations were considered to be reliable estimates of the number of live births so he used the total number of births over a three year period. The sample estimate was 71,866.333 baptisms per year over this period giving a ratio of one registered baptism for every 28.35 persons. The total number of baptismal registrations for France was also available to him and he assumed that the ratio of live births to population was constant. He then used the ratio from his sample to estimate the population of France.

[[Karl Pearson]] noted in 1897 that the ratio estimates are biased and cautioned against their use.<ref name=Pearson1897>Pearson K (1897) On a form of spurious correlation that may arise when indices are used for the measurement of organs. Proc Roy Soc Lond 60: 498</ref>

==See also==

*[[Ratio distribution]]

==References==
{{reflist}}

{{Statistics|descriptive|state=collapsed}}

[[Category:Statistical deviation and dispersion]]
[[Category:Articles containing proofs]]
[[Category:Data analysis]]
[[Category:Statistical ratios]]