6.3 Diagnostics and Hypothesis Testing

6.3 Diagnostics and Hypothesis Testing

Analysis of Residuals of the Model

The residuals carry important information concerning the appropriation of assumption. Analyses may include informal graphics to display general features of the residual as well as formal tests to detect specific departure from underlying assumptions. 

Each formal and informal procedure is complementary and both have a place in residual analysis. 

The residual models are also called Measurement Error Models. For simple Linear Measurement Error model, the goal is to estimate a straight line fit between two variables from bivariate data, both of which are measured with error. 

The standard linear regression model iswhere, X is matrix of known constants. Y is an n-vector of observable response. β is an unknown parameter. ε is an unobservable error with indicated distributional properties. To asses the appropriateness of model, for a given problem, it is necessary to determine if the assumptions about errors are reasonable. Since errors ε are unobservable, this must be done indirectly using residuals. In regression model, it is assumed that the error term satisfies:
Standard Residual Plots

Residuals can be used in a variety of graphical and non graphical summaries to identify in approximate assumptions. Generally, a number of different plots will be required to extract the available information. Standard Residuals plots are those in which ei are plotted against fitted values ?i. The plots are commonly used to diagnose non linearity and non constant error variance. These plots give relevant information about the fit of the model.
Identifiability of the Model 

One of the most important differences between residual models and ordinary regression models is concerned with model identifiability. It is common to assume that all random variables in the residual model are jointly normal. This means that different sets of parameters can lead to the same joint distributions of x and y. In this situation, it is impossible to estimate consistently the parameters from the data. 

B. Hypothesis Testing

1. Fundamentals Concepts of Hypothesis Testing

A Hypothesis may be defined as a statement about one or more populations. The hypothesis is frequently concerned with parameters of the populations about which the statement is made.

Hypothesis testing is of considerable importance in Six Sigma. The hypothesis test is designed to make an inference about the true population value at a significant level of confidence. 

For example: A hospital administrator may hypothesize that the average length of stay of patients admitted to the hospitals is five days or a physician may hypothesize that a certain drug will be effective in 90 % of cases for which it is used. By means of hypothesis testing one determines whether or not such statements are compatible with the available data.

a. Hypothesis Testing

1. Study the data
The nature of the data that form the basis of the testing procedures must be understood, since this determines the particular test to be employed. Whether the data consists of counts or measurements must be determined. 

2. Set up a hypothesis
The first thing in hypothesis testing is to set up a hypothesis about a population parameter. There are two statistical hypothesis involved in hypotheses testing. 

The null hypothesis is the hypothesis to be tested designated by symbol H 0. This is referred to as a hypothesis of no difference. In testing processes, the null hypothesis is either rejected or is not rejected. If the null hypothesis is not rejected, this means that the data on which the test is based do not provide sufficient evidence to cause rejection. If the testing procedure leads to rejection, it is said that the data in hand is not compatible with the null hypothesis. 

The alternative hypothesis is a statement of what is believed is true if the sample data causes you to reject the null hypothesis. It is designated by H A. The null and alternative hypotheses are complementary.

For example, A psychologist who wishes to test whether or not a certain class of people have a mean I.Q. higher than 100 might establish the following null and alternatives hypotheses:2. Test statistic
The test statistic is some statistic that may be computed from the data of the sample. There are many possible values that the test statistic may assume. The test statistic serves as a decision maker, since the decision is to reject or not to reject the null hypothesis depends on the magnitude of the test statistic. In general,For example, A test statistic used for large sample size in a continuous normal distribution isThe distribution of the test statistic follows standard normal distribution if the null hypothesis is true. 

3. Decision Rule
The decision rule tells about the rejection of the null hypothesis if the value of the test statistic that is computed from the sample is one of the values in the rejection region (region where values that are less likely to occur if the null hypothesis is true), and not to reject the null hypothesis if the computed value of the test statistic is one of the values in the non rejection region. 

b. Significance Level, Type I and Type II Errors, Power

The level of significance, designated by α is a probability and, in fact, is the probability of rejecting a true null hypothesis.

The decision as to which values go into the rejection region and which ones go into the non-rejection region is made on the basis of the desired level of significance, α. The term reflects the fact that hypothesis tests are sometimes called significance tests, and the computed value of the test statistic that falls in the rejection region is called ‘significant’. 

A small value of α is selected in order to make the probability of rejecting a true null hypothesis small. The more frequently encountered values of α are .01, .05 and .10. 

The following diagram illustrates the region in which one would accept or reject the null hypothesis when it is being tested at 5 percent level significance and a two-tail is employed. It may be noted that 2.5 percent of the area under the curve is located in each tail.Types of Errors

When a statistical hypothesis is tested there are four possibilities: 

1. The hypothesis is true but your test rejects it. (Type I error) 
2. The hypothesis is false but your test accepts it. (Type II error) 
3. The hypothesis is true but your test accepts it. (Correct Decision) 
4. The hypothesis is false but your test rejects it. (Correct Decision)For example:

Assume that the difference between two population means is actually zero. If the test of significance when applied to the sample means gives that the difference in population means is significant, you make a Type I Error. On the other hand, suppose there is a true difference between the two population means. Now if the test of significance leads to the judgment “not significant”, you commit a Type II Error.

It is more dangerous to accept a false hypothesis (Type II Error) than to reject a correct one (Type I Error). While testing hypothesis the aim should be to reduce both the errors. But due to fixed sample size, it is not possible to control both the errors simultaneously. Hence you keep the probability of committing Type I Error at a certain fixed level, called level of significance and then reduce Type II Error to get better results.