Ratio Estimator in Stratified Sampling
Ratio Estimator in Stratified Sampling
- Rohan Regi (2148111)
Introduction:
The ratio
estimator is a statistical parameter that equals the ratio of two random
variables' means. When ratio estimates are utilized in experimental or survey
work, they must be corrected for bias. Because the ratio estimates are
asymmetrical, symmetrical tests like the t test should be avoided when
generating confidence intervals.
Under
various scenarios, the ratio estimator was found to be more exact than the
traditional sample mean estimator in calculating the population mean of the
studied character. Several academics have shifted their focus to finding more
exact estimates by exploiting the prior value of specific demographic factors.
At the estimate step, Searls (1964) employed the coefficient of variation of the
studied character. The coefficient of variation is rarely known in practice.
Various authors, including Sen (1978), Sesodiya, and Dewivedi, were inspired by
Searls' (1964) work (1981) In the ratio technique of estimation, Singh et al
(1991) and Upadhyaya and Singh (1984) employed the known coefficient of
variation of an auxiliary character to estimate the population mean of the
study character. Singh et al. (1973) were the first to employ the previous
value of the coefficient of kurtosis in calculating the study character's
population variance. Searls and Interapanich later utilized it (1990). Singh
and Tailor (2003) have suggested a modified ratio estimator based on the known
correlation coefficient value. When priori information on an auxiliary variable
with some attribute is known, Jhajj et al (2006) and Singh, et al (2008)
defined ratio estimators of population mean using the point biserial
correlation coefficient between auxiliary attribute and study variable.
In stratified random
sampling with auxiliary attributes, certain ratio-type estimators have been
developed. Up to first level of approximation, equations for the bias and mean
square errors of the proposed estimators have been developed. The suggested
estimators are proven to be more efficient than the traditional combined ratio
estimator under specific conditions when compared to the traditional combined
ratio estimator.
There are two techniques
to create estimates when utilizing ratio estimation with stratified random sampling.
One method is to estimate ratios independently in each stratum before combining
them. This creates a separate ratio estimator. The second method is to utilize estimators for stratified random sampling to calculate estimators for
µy and µx, and then use y(bar)st/x(bar)st as a ratio estimator of µy/µx. A
combined ratio estimator is the result of this.
Separate
ratio-type estimators for the population mean
are studied, along with their features. Separate ratio-type estimators for
population mean based on known auxiliary variate parameters are proposed. The
proposed estimators' bias and mean squared error are calculated up to the first
degree of approximation. Under particular given circumstances, the suggested
estimators are demonstrated to be more efficient than unbiased estimators in
stratified random sampling and typical separate ratio estimators.
Combined ratio estimator: In the case of a separate estimator, it is assumed that the nm's in each stratum were big. However, it may not always be true in practice. X_bar is the population mean of X based on all the N = ∑_(i=1)^(N ) [N_i ] units. It does not depend on individual stratum units. It does not depend on information on each Xi but only on X.
The separate ratio estimator's main drawback is that with
small sample sizes per stratum, the individual stratum variance estimates will
be biased, and this bias will be increased across strata. Unless the stratum
sizes are tiny, such as (ni < 20), or the within-stratum ratios are
almost identical, it is advised to employ the separate ratio estimator. The
population totals are estimated by multiplying by the population number N,
resulting in 〖τy〗_(RS )= N〖µy〗_(RS ) or 〖τy〗_(RC )
= N〖µy〗_(RC )
Formulas:
Estimating the Ratio
In a situation with two strata (labelled A and B), the expression for estimating a mean using the separate ratio estimator is
In a situation with two strata (labelled A and B), the expression for estimating a mean using the combined ratio estimator is:
with estimated variance:
Properties of separate ratio estimator:
Properties of combined ratio estimator:
Comparison of combined and separate ratio estimators
An obvious question arises that which of the estimates (^Y_RS) or (^Y_RC) is better. So, we compare their MSEs. Note that the only difference in the term of these MSEs is due to the form of ratio estimate. It is
The difference D depends on
(i)
The
magnitude of the difference between the strata ratios (R_i) and whole population ratio (R).
(ii) The value of R_i S_ix^2- rS_ix S_iy is usually small and vanishes when the regression line of y on x is linear and passes through origin within each stratum. In such a case
Advantages and Disadvantages of Ratio Estimator in stratified sampling:
Advantages
1. Helps
in forecasting and planning by performing trend analysis.
2. Helps
in estimating budget for the firm by analyzing previous trends.
3. It
helps in determining how efficiently a firm or an organization is operating.
4. It
provides significant information to users of accounting information regarding
the performance of the business.
5. It
helps in comparison of two or more firms.
6. It
helps in determining both liquidity and long-term solvency of the firm.
Disadvantages:
1.
Financial statements seem to be
complicated.
2.
Several organizations work in various
enterprises each possessing different environmental positions such as market
structure, regulation, etc., Such factors are important that a comparison of 2 organizations from varied industries might be ambiguous.
3.
Financial accounting data is
influenced by views and hypotheses. Accounting criteria provide different
accounting methods, which reduces comparability and thus ratio analysis is less
helpful in such circumstances.
4.
Ratio analysis illustrates the
associations between prior data while users are more concerned about current
and future data.
Application:
Cardiovascular diseases (CVDs) are the
number 1 cause of death globally, taking an estimated 17.9 million lives each
year, which accounts for 31% of all deaths worldwide. Four out of 5CVD deaths
are due to heart attacks and strokes, and one-third of these deaths occur
prematurely in people under 70 years of age. Heart failure is a common event caused
by CVDs and this dataset contains 11 features that can be used to predict a
possible heart disease.
People
with cardiovascular disease or who are at high cardiovascular risk (due to the
presence of one or more risk factors such as hypertension, diabetes,
hyperlipidemia or already established disease) need early detection and
management wherein a machine learning model can be of great help.
Objective:
Here, our objective is to
find the ratio estimates with stratified sampling for this particular dataset
for variables of interest as serum cholesterol
[mm/dl] and maximum heart rate achieved [Numeric value between 60 and 202].
Attribute
Information
- Age:
age of the patient [years]
- Sex:
sex of the patient [M: Male, F: Female]
- ChestPainType:
chest pain type [TA: Typical Angina, ATA: Atypical Angina, NAP:
Non-Anginal Pain, ASY: Asymptomatic]
- RestingBP:
resting blood pressure [mm Hg]
- Cholesterol:
serum cholesterol [mm/dl]
- FastingBS:
fasting blood sugar [1: if FastingBS > 120 mg/dl, 0: otherwise]
- RestingECG:
resting electrocardiogram results [Normal: Normal, ST: having ST-T wave
abnormality (T wave inversions and/or ST elevation or depression of >
0.05 mV), LVH: showing probable or definite left ventricular hypertrophy
by Estes' criteria]
- MaxHR:
maximum heart rate achieved [Numeric value between 60 and 202]
- ExerciseAngina:
exercise-induced angina [Y: Yes, N: No]
- Oldpeak:
oldpeak = ST [Numeric value measured in depression]
- ST_Slope:
the slope of the peak exercise ST segment [Up: upsloping, Flat: flat,
Down: downsloping]
- HeartDisease:
output class [1: heart disease, 0: Normal]
Source
This
dataset was created by combining different datasets already available
independently but not combined before. In this dataset, 5 heart datasets are
combined over 11 common features which makes it the largest heart disease
dataset available so far for research purposes. The five datasets used for its
curation are:
- Cleveland:
303 observations
- Hungarian:
294 observations
- Switzerland:
123 observations
- Long
Beach VA: 200 observations
- Stalog
(Heart) Data Set: 270 observations
Total:
1190 observations
Duplicated: 272 observations
Final
dataset: 918 observations
Every
dataset used can be found under the Index of heart disease datasets from UCI
Machine Learning Repository on the following link: https://archive.ics.uci.edu/ml/machine-learning-databases/heart-disease/
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