GTU MCA III (SM): November 2010

Sunday, November 28, 2010

Regression (UNIT -5)

In statistics, regression analysis includes any techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables.

History

The earliest form of regression was the method of least squares (French: méthode des moindres carrés), which was published by Legendre in 1805, and by Gauss in 1809.Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun. Gauss published a further development of the theory of least squares in 1821, including a version of the Gauss–Markov theorem.
The term "regression" was coined by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean). For Galton, regression had only this biological meaning, but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context.. In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925 . Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.
Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than observations, and causal inference with regression.

Regression models

Regression models involve the following variables:

The unknown parameters denoted as β; this may be a scalar or a vector.
The independent variables, X.
The dependent variable, Y.

In various fields of application, different terminologies are used in place of dependent and independent variables.
A regression model relates Y to a function of X and β.

$Y \approx f (\mathbf {X}, \boldsymbol{\beta} )$

The approximation is usually formalized as E(Y | X) = f(X, β). To carry out regression analysis, the form of the function f must be specified. Sometimes the form of this function is based on knowledge about the relationship between Y and X that does not rely on the data. If no such knowledge is available, a flexible or convenient form for f is chosen.
Assume now that the vector of unknown parameters β is of length k. In order to perform a regression analysis the user must provide information about the dependent variable Y:

If N data points of the form (Y,X) are observed, where N < k, most classical approaches to regression analysis cannot be performed: since the system of equations defining the regression model is underdetermined, there is not enough data to recover β.
If exactly N = k data points are observed, and the function f is linear, the equations Y = f(X, β) can be solved exactly rather than approximately. This reduces to solving a set of N equations with N unknowns (the elements of β), which has a unique solution as long as the X are linearly independent. If f is nonlinear, a solution may not exist, or many solutions may exist.
The most common situation is where N > k data points are observed. In this case, there is enough information in the data to estimate a unique value for β that best fits the data in some sense, and the regression model when applied to the data can be viewed as an overdetermined system in β

In the last case, the regression analysis provides the tools for:

Finding a solution for unknown parameters β that will, for example, minimize the distance between the measured and predicted values of the dependent variable Y (also known as method of least squares).
Under certain statistical assumptions, the regression analysis uses the surplus of information to provide statistical information about the unknown parameters β and predicted values of the dependent variable Y.

Necessary number of independent measurements

Consider a regression model which has three unknown parameters, β₀, β₁, and β₂. Suppose an experimenter performs 10 measurements all at exactly the same value of independent variable vector X (which contains the independent variables X₁, X₂, and X₃). In this case, regression analysis fails to give a unique set of estimated values for the three unknown parameters; the experimenter did not provide enough information. The best one can do is to estimate the average value and the standard deviation of the dependent variable Y. Similarly, measuring at two different values of X would give enough data for a regression with two unknowns, but not for three or more unknowns.

If the experimenter had performed measurements at three different values of the independent variable vector X, then regression analysis would provide a unique set of estimates for the three unknown parameters in β.
In the case of general linear regression, the above statement is equivalent to the requirement that matrix X^TX is invertible.

Statistical assumptions

When the number of measurements, N, is larger than the number of unknown parameters, k, and the measurement errors ε_i are normally distributed then the excess of information contained in (N - k) measurements is used to make statistical predictions about the unknown parameters. This excess of information is referred to as the degrees of freedom of the regression.

Underlying assumptions

Classical assumptions for regression analysis include:

The sample is representative of the population for the inference prediction.
The error is a random variable with a mean of zero conditional on the explanatory variables.
The independent variables are measured with no error. (Note: If this is not so, modeling may be done instead using errors-in-variables model techniques).
The predictors are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others. See Multicollinearity.
The errors are uncorrelated, that is, the variance-covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
The variance of the error is constant across observations (homoscedasticity). (Note: If not, weighted least squares or other methods might instead be used).

Linear regression

In linear regression, the model specification is that the dependent variable,

y i

is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling

n

data points there is one independent variable:

x i

, and two parameters,

β 0

and

β 1

straight line: $y_i=\beta_0 +\beta_1 x_i +\varepsilon_i,\quad i=1,\dots,n.\!$

In multiple linear regression, there are several independent variables or functions of independent variables. For example, adding a term in x_i² to the preceding regression gives:

parabola: $y_i=\beta_0 +\beta_1 x_i +\beta_2 x_i^2+\varepsilon_i,\ i=1,\dots,n.\!$

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable

x i

, it is linear in the parameters

β 0

β 1

and

β 2 .

In both cases, $\varepsilon_i$ is an error term and the subscript

i

indexes a particular observation. Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

$\widehat{y_i} = \widehat{\beta}_0 + \widehat{\beta}_1 x_i.$

The residual, $e_i = y_i - \widehat{y}_i$ , is the difference between the value of the dependent variable predicted by the model, $\widehat{y_i}$ and the true value of the dependent variable

y i

. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSE, also sometimes denoted RSS:

$SSE=\sum_{i=1}^N e_i^2. \,$

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, $\widehat{\beta}_0, \widehat{\beta}_1$ .

Illustration of linear regression on a data set.

In the case of simple regression, the formulas for the least squares estimates are

$\widehat{\beta_1}=\frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}\text{ and }\hat{\beta_0}=\bar{y}-\widehat{\beta_1}\bar{x}$

where $\bar{x}$ is the mean (average) of the

x

values and $\bar{y}$ is the mean of the

y

values. See simple linear regression for a derivation of these formulas and a numerical example. Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:

$\hat{\sigma}^2_\varepsilon = \frac{SSE}{N-2}.\,$

This is called the mean square error (MSE) of the regression. The standard errors of the parameter estimates are given by

$\hat\sigma_{\beta_0}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{N} + \frac{\bar{x}^2}{\sum(x_i-\bar x)^2}}$

$\hat\sigma_{\beta_1}=\hat\sigma_{\varepsilon} \sqrt{\frac{1}{\sum(x_i-\bar x)^2}}.$

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

General linear model

In the more general multiple regression model, there are p independent variables:

$y_i = \beta_0 + \beta_1 x_{1i} + \cdots + \beta_p x_{pi} + \varepsilon_i, \,$

The least square parameter estimates are obtained by p normal equations. The residual can be written as

$e_i=y_i - \hat\beta_0 - \hat\beta_1 x_1 - \cdots - \hat\beta_p x_p.$

The normal equations are

$\sum_{i=1}^n \sum_{k=1}^p X_{ij}X_{ik}\hat \beta_k=\sum_{i=1}^n X_{ij}y_i,\ j=1,\dots,p.\,$

Note that for the normal equations depicted above, $y_i = \beta_1 x_{1i} + \cdots + \beta_p x_{pi} + \varepsilon_i \,$
That is, there is no

β 0

. Thus in what follows, $\boldsymbol \beta = (\beta_1, \beta_2, \dots, \beta_p).$
In matrix notation, the normal equations for

k

responses (usually

k = 1

) are written as

$\mathbf{_p(X_n^\top X )_p\hat \boldsymbol \beta_k= _pX_n^\top Y_k}.\,$

with generalized inverse (

-

) solution, subscripts showing matrix dimensions:

$\mathbf{_p\hat \boldsymbol \beta_k= {}_p(X_n^\top X )_p^-X_n^\top Y_k}.\,$

Nonlinear regression

When the model function is not linear in the parameters, the sum of squares must be minimized by an iterative procedure.

Wednesday, November 24, 2010

Chi-Square(Unit -4)

Pearson's chi-square is used to assess two types of comparison: tests of goodness of fit and tests of independence. A test of goodness of fit establishes whether or not an observed frequency distribution differs from a theoretical distribution. A test of independence assesses whether paired observations on two variables, expressed in a contingency table, are independent of each other – for example, whether people from different regions differ in the frequency with which they report that they support a political candidate.

The first step in the chi-square test is to calculate the chi-square statistic. In order to avoid ambiguity, the value of the test-statistic is denoted by Χ² rather than χ² (i.e. uppercase chi instead of lowercase); this also serves as a reminder that the distribution of the test statistic is not exactly that of a chi-square random variable. However some authors do use the χ² notation for the test statistic. An exact test which does not rely on using the approximate χ² distribution is Fisher's exact test: this is significantly more accurate in evaluating the significance level of the test, especially with small numbers of observation.

The chi-square statistic is calculated by finding the difference between each observed and theoretical frequency for each possible outcome, squaring them, dividing each by the theoretical frequency, and taking the sum of the results. A second important part of determining the test statistic is to define the degrees of freedom of the test: this is essentially the number of observed frequencies adjusted for the effect of using some of those observations to define the "theoretical frequencies".

The chi-square (I) test is used to determine whether there is a significant difference between the expected frequencies and the observed frequencies in one or more categories. Do the number of individuals or objects that fall in each category differ significantly from the number you would expect? Is this difference between the expected and observed due to sampling error, or is it a real difference?

Ø Chi-Square Test Requirements

1. Quantitative data.

2. One or more categories.

3. Independent observations.

4. Adequate sample size (at least 10).

5. Simple random sample.

6. Data in frequency form.

7. All observations must be used.

Ø Expected Frequencies

When you find the value for chi square, you determine whether the observed frequencies differ significantly from the expected frequencies. You find the expected frequencies for chi square in three ways:

1 . You hypothesize that all the frequencies are equal in each category. For example, you might expect that half of the entering freshmen class of 200 at Tech College will be identified as women and half as men. You figure the expected frequency by dividing the number in the sample by the number of categories. In this example, where there are 200 entering freshmen and two categories, male and female, you divide your sample of 200 by 2, the number of categories, to get 100 (expected frequencies) in each category.

2. You determine the expected frequencies on the basis of some prior knowledge. Let's use the Tech College example again, but this time pretend we have prior knowledge of the frequencies of men and women in each category from last year's entering class, when 60% of the freshmen were men and 40% were women. This year you might expect that 60% of the total would be men and 40% would be women. You find the expected frequencies by multiplying the sample size by each of the hypothesized population proportions. If the freshmen total were 200, you would expect 120 to be men (60% x 200) and 80 to be women (40% x 200).Now let's take a situation, find the expected frequencies, and use the chi-square test to solve the problem.

Ø Situation

Thai, the manager of a car dealership, did not want to stock cars that were bought less frequently because of their unpopular color. The five colors that he ordered were red, yellow, green, blue, and white. According to That, the expected frequencies or number of customers choosing each color should follow the percentages of last year. She felt 20% would choose yellow, 30% would choose red, 10% would choose green, 10% would choose blue, and 30% would choose white. She now took a random sample of 150 customers and asked them their color preferences. The results of this poll are shown in Table 1 under the column labeled _observed frequencies."

Table 1 - Color Preference for 150 Customers for Thai's Superior Car Dealership

Category Color	Observed Frequencies	Expected Frequencies
Yellow	35	30
Red	50	45
Green	30	15
Blue	10	15
White	25	45

last year, we would expect 20% to choose yellow. Figure the expected frequencies for yellow by taking 20% of the 150 customers, getting an expected frequency of 30 people for this category. For the color red we would expect 30% out of 150 or 45 people to fall in this category. Using this method, Thai figured out the expected frequencies 30, 45, 15, 15, and 45. Obviously, there are discrepancies between the colors preferred by customers in the poll taken by Thai and the colors preferred by the customers who bought their cars last year. Most striking is the difference in the green and white colors. If Thai were to follow the results of her poll, she would stock twice as many green cars than if she were to follow the customer color preference for green based on last year's sales.In the case of white cars, she would stock half as many this year. What to do??? Thai needs to know whether or not the discrepancies between last year's choices (expected frequencies) and this year's preferences on the basis of his poll (observed frequencies) demonstrate a real change in customer color preferences. It could be that the differences are simply a result of the random sample she chanced to select. If so, then the population of customers really has not changed from last year as far as color preferences go. The null hypothesis states that there is no significant difference between the expected and observed frequencies. The alternative hypothesis states they are different. The level of significance (the point at which you can say with 95% confidence that the difference is NOT due to chance alone) is set at .05 (the standard for most science experiments.) The chi-square

formula used on these data is

X2 = (O - E)2 where O is the Observed Frequency in each category

E E is the Expected Frequency in the corresponding category

df is the "degree of freedom" (n-1)

X2 is Chi Square

Ø PROCEDURE

We are now ready to use our formula for X2 and find out if there is a significant difference between the observed and expected frequencies for the customers in choosing cars. We will set up a worksheet; then you will follow the directions to form the columns and solve the formula.

2. After calculating the Chi Square value, find the "Degrees of Freedom." (DO NOT SQUARE THE NUMBER YOU GET, NOR FIND THE SQUARE ROOT – THE NUMBER YOU GET FROM COMPLETING THE CALCULATIONS AS ABOVE IS _CHI SQUARE.)

Degrees of freedom (df) refers to the number of values that are free to vary after restriction has been placed on the data. For instance, if you have four numbers with the restriction that their sum has to be 50, and then three of these numbers can be anything, they are free to vary, but the fourth number definitely is restricted. For example, the first three numbers could be 15, 20, and 5, adding up to 40; then the fourth number has to be 10 in order that they sum to 50. The degrees of freedom for these values are then three. The degrees of freedom here is defined as N - 1, the number in the group minus one restriction (4 - I ).

3. Find the table value for Chi Square. Begin by finding the df found in step 2 along the left hand side of the table. Run your fingers across the proper row until you reach the predetermined level of significance (.05) at the column heading on the top of the table. The table value for Chi Square in the correct box of 4 df and P=.05 level of significance is 9.49.

4. If the calculated chi-square value for the set of data you are analyzing (26.95) is equal to or greater than the table value (9.49 ), reject the null hypothesis. There IS a significant difference between the data sets that cannot be due to chance alone. If the number you calculate is LESS than the number you find on the table, than you can probably say that any differences are due to chance alone.

In this situation, the rejection of the null hypothesis means that the differences between the expected frequencies (based upon last year's car sales) and the observed frequencies (based upon this year's poll taken by Thai) are not due to chance. That is, they are not due to chance variation in the sample Thai took; there is a real difference between them. Therefore, in deciding what color autos to stock, it would be to Thai's advantage to pay careful attention to the results of her poll!

Ø The steps in using the chi-square test may be summarized as follows:

ü Chi-square

I. Write the observed frequencies in column O

2. Figure the expected frequencies and write them in column E.

3. Use the formula to find the chi-square value:

4. Find the df. (N-1)

5. Find the table value (consult the Chi Square Table.)

6. If your chi-square value is equal to or greater than the table value,

reject the null

Hypothesis: differences in your data are not due to chance alone

Ø Problem

The approximation to the chi-square distribution breaks down if expected frequencies are too low. It will normally be acceptable so long as no more than 20% of the events have expected frequencies below 5. Where there is only 1 degree of freedom, the approximation is not reliable if expected frequencies are below 10. In this case, a better approximation can be obtained by reducing the absolute value of each difference between observed and expected frequencies by 0.5 before squaring; this is called Yates' correction for continuity.

In cases where the expected value, E, is found to be small (indicating either a small underlying population probability, or a small number of observations), the normal approximation of the multinomial distribution can fail, and in such cases it is found to be more appropriate to use the G-test, a likelihood ratio-based test statistic. Where the total sample size is small, it is necessary to use an appropriate exact test, typically either the binomial test or (for contingency tables) Fisher's exact test; but note that this test assumes fixed and known marginal totals.

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