# Design of Experiments Overview

An experiment is usually a test or trial or series of tests. The objective of the experiment may either be

1. Confirmation
2. Exploration

Designing of an experiment means, providing a plan and actual procedure of laying out the experiment. It is a design of any information gathering exercise where variation is present under the full or no control of the experimenter. The experimenter in design of experiments is often interested in the effect of some process or intervention (the treatment) on some objects (the experimental units) such as people, parts of people, groups of people, plants, animals etc. So the design of experiment is an efficient procedure for planning experiments so that the data obtained can be analyzed to yield and object conclusions.

In observational study the researchers observe individuals and measure variables of interest but do not attempt to influence the response variable, while in an experimental study, the researchers deliberately (purposely) impose some treatment on individuals and then observe the response variables. When the goal is to demonstrate cause and effect, experiment is the only source of convincing data.

## Statistical Design

By the statistical design of experiments, we refer to the process of planning the experiment, so that the appropriate data will be collected, which may be analyzed by statistical methods resulting in valid and objective conclusions. Thus there are two aspects to any experimental problem:

1. The design of the experiments
2. The statistical analysis of the data

There are many experimental design which differ from each other primarily in the way, in which the experimental units are classified, before the application of treatment.

## Design of experiment (DOE) helps in

• Identifying the relationships between cause and effect
• Provide some understanding of interactions among causative factors
• Determining the level at which to set the controllable factors in order to optimize reliability
• Minimizing the experimental error i.e., noise
• Improving the robustness of the design or process to variation

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# The Level of Measurements

In statistics, data can be classified according to level of measurement, dictating the calculations that can be done to summarize and present the data (graphically), it also helps to determine, what statistical tests should be performed. For example, suppose there are six colors of candies in a bag and you assign different numbers (codes) to them in such a way that brown candy has a value of 1, yellow 2, green 3, orange 4, blue 5, and red a value of 6. From this bag of candies, adding all the assigned color values and then dividing by the number of candies, yield an average value of 3.68. Does this mean that the average color is green or orange? Of course not. When computing statistic, it is important to recognize the data type, which may be qualitative (nominal and ordinal) and quantitative (Interval and ratio).

The level of measurements has been developed in conjunction with the concepts of numbers and units of measurement. Statisticians classified measurements according to levels. There are four level of measurements, namely, nominal, ordinal, interval and ratio, described below.

Nominal Level of Measurement

In nominal level of measurement, the observation of a qualitative variable can only be classified and counted. There is no particular order to the categories. Mode, frequency table, pie chart and bar graph are usually drawn for this level of measurement.

Ordinal Level of Measurement

In ordinal level of measurement, data classification are presented by sets of labels or names that have relative values (ranking or ordering of values). For example, if you survey 1,000 people and ask them to rate a restaurant on a scale ranging from 0 to 5, where 5 shows higher score (highest liking level) and zero shows the lowest (lowest liking level). Taking the average of these 1,000 people’s response will have meaning. Usually graphs and charts are drawn for ordinal data.

Interval Level of Measurement

Numbers also used to express the quantities, such as temperature, dress size and plane ticket are all quantities. The interval level of measurement allows for the degree of difference between items but no the ratio between them. There is meaningful difference between values, for example 10 degrees Fahrenheit and 15 degrees is 5, and the difference between 50 and 55 degrees is also 5 degrees. It is also important that zero is just a point on the scale, it does not represents the absence of heat, just that it is freezing point.

Ratio Level of Measurement

All of the quantitative data is recorded on the ratio level. It has all the characteristics of the interval level, but in addition, the zero point is meaningful and the ratio between two numbers is meaningful. Examples of ratio level are wages, units of production, weight, changes in stock prices, distance between home and office, height etc.
Many of the inferential test statistics depends on ratio and interval level of measurement. Many author argue that interval and ratio measures should be named as scale.

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# p-value interpretation, definition, introduction and examples

The p-value also known as observed level of significance or exact level of significance or the exact probability of committing a type-I error (probability of rejecting H0, when it is true), helps to determine the significance of results from hypothesis. The p-value is the probability of obtaining the observed sample results or a more extreme result when the null hypothesis (a statement about population) is actually true.

In technical words, one can define p-value as the lowest level of significance at which a null hypothesis can be rejected. If p-value is very small or less than the threshold value (chosen level of significance), then the observed data is considered as inconsistent with the assumption that the null hypothesis is true and thus null hypothesis must be rejected while the alternative hypothesis should be accepted. The p-value is a number between 0 and 1 and in literature it is usually interpreted in the following way:

• A small p-value (<0.05) indicates strong evidence against the null hypothesis
• A large p-value (>0.05) indicates weak evidence against the null hypothesis.
• p-value very close to the cutoff (say 0.05) are considered to be marginal.

Let the p-value of a certain test statistic is 0.002 then it means that the probability of committing a type-I error (making a wrong decision) is about 0.2 percent, that is only about 2 in 1,000. For a given sample size, as | t | (or any test statistic) increases the p-value decreases, so one can reject the null hypothesis with increasing confidence.

Fixing the level of significance ($\alpha$) (i.e. type-I error) equal to the p-value of a test statistic then there is no conflict between the two values, in other words, it is better to give up fixing up (significance level) arbitrary at some level of significance such as (5%, 10% etc.) and simply choose the p-value of the test statistic. For example, if the p-value of test statistic is about 0.145 then one can reject the null hypothesis at this exact significance level as nothing wrong with taking a chance of being wrong 14.5% of the time of someone reject the null hypothesis.

p-value addresses only one question: how likely are your data, assuming a true null hypothesis? It  does not measure support for the alternative hypothesis.

Most authors refers to p-value<0.05 as statistically significant and p-value<0.001 as highly statistically significant (less than one in a thousand chance of being wrong).

p-value is usually incorrectly interpreted as it is usually interpreted as the probability of making a mistake by rejecting a true null hypothesis (a Type-I error). p-value cannot be error rate because:

p-value is calculated based on the assumption that the null hypothesis is true and that the difference in the sample by random chances. Consequently, p-value cannot tell about the probability that the null hypothesis is true or false because it is 100% true from the perspective of the calculations.

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# Degrees of Freedom

The degrees of freedom (df) or number of degrees of freedom refers to the number of observations in a sample minus the number of (population) parameters being estimated from the sample data. All this means that the degrees of freedom is a function of both sample size and the number of independent variables. In other words it is the number of independent observations out of a total of ($n$) observations.

In statistics, the degrees of freedom considered as the number of values in a study that are free to vary. For example (degrees of freedom example in real life), if you have to take ten different courses to graduate, and only ten different courses are offered, then you have nine degrees of freedom. Nine semesters you will be able to choose which class to take; the tenth semester, there will only be one class left to take – there is no choice, if you want to graduate, this is the concept of the degrees of freedom (df) in statistics.

Let a random sample of size n is taken from a population with an unknown mean $\overline{X}$. The sum of the deviations from their means is always equal to zero i.e.$\sum_{i=1}^n (X_i-\overline{X})=0$. This require a constraint on each deviation $X_i-\overline{X}$ used when calculating the variance.

$S^2 =\frac{\sum_{i=1}^n (X_i-\overline{X}) }{n-1}$

This constraint (restriction) implies that $n-1$ deviations completely determine the nth deviation. The $n$ deviations (and also the sum of their squares and the variance in the $S^2$ of the sample) therefore $n-1$ degrees of freedom.

A common way to think of degrees of freedom is as the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a parameter of the population from which that sample is drawn. For example, if we have two observations, when calculating the mean we have two independent observations; however, when calculating the variance, we have only one independent observation, since the two observations are equally distant from the mean.

Single sample: For $n$ observation one parameter (mean) needs to be estimated, that leaves $n-1$ degrees of freedom for estimating variability (dispersion).

Two samples: There are total of $n_1+n_2$ observations ($n_1$ for group1 and $n_2$ for group2) and two means need to be estimated, which leaves $n_1+n_2-2$ degrees of freedom for estimating variability.

Regression with p predictors: There are $n$ observations with $p+1$ parameters needs to be estimated (regression coefficient for each predictor and the intercept). This leaves $n-p-1$ degrees of freedom of error, which accounts for the error degrees of freedom in the ANOVA table.

Several commonly encountered statistical distributions (Student’s t, Chi-Squared, F) have parameters that are commonly referred to as degrees of freedom. This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector. If $X_i; i=1,2,\cdots, n$ are independent normal $(\mu, \sigma^2)$ random variables, the statistic (formula) is $\frac{\sum_{i=1}^n (X_i-\overline{X})^2}{\sigma^2}$, follows a chi-squared distribution with $n-1$ degrees of freedom. Here, the degrees of freedom arises from the residual sum of squares in the numerator and in turn the $n-1$ degrees of freedom of the underlying residual vector ${X_i-\overline{X}}$.

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# Binary Logistic Regression Minitab Tutorial

Binary Logistic Regression is used to perform logistic regression on a binary response (dependent) variable (a variable only that has two possible values, such as presence or absence of a particular disease, this kind of variable is known as dichotomous variable i.e binary in nature).

Binary Logistic Regression can classify observations into one of two categories. These classifications can give fewer classification errors than discriminant analysis for some cases.

The default model contains the variables that you enter in Continuous predictors and Categorical predictors. You can also add interaction and/or polynomial terms by using the tools available in model sub-dialog box.

Minitab stores the last model that you fit for each response variable. This stored models can be used to quickly generate predictions, contour plots, surface plots, overlaid contour plots, factorial plots, and optimized responses.

To perform a Binary Logistic Regression Analysis in Minitab, follow the steps given below. It is assumed that you have already launched the Minitab software.

Step1:  Choose Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model.

Fit Binary Logistic Regression

Step2:  Do one of the following:

If your data is in raw or frequency form, follow these steps:

Response in Binary Logistic Regression (Frequency Format)

1. Choose Response in binary response/frequency format, from combobox on top
2. In Response text box, enter the column that contains the response variable.
3. In Frequency text box, enter the optional column that contains the count or frequency variable.

If you have summarized data, then follow these steps:

Response in Binary Logistic Regression (Trial Format)

1. Choose Response in event/trial format, from combobox on top of the dialog box.
2. In Number of events, enter the column that contains the number of times the event occurred in your sample at each combination of the predictor values.
3. In Number of trials, enter the column that contains the corresponding number of trials.

Step4:  In Continuous predictors, enter the columns that contain continuous predictors. In Categorical predictors, enter the columns that contain categorical predictors. You can add interactions and other higher order terms to the model.

Step5:  If you like, use one or more of the dialog box options, then click OK.

The following are options available in the main dialog box of Minitab Binary Logistic Regression:

Response in binary response/frequency format: Choose if the response data has been entered as a column that contains 2 distinct values i.e as a dichotomous variable.
Response: Enter the column that contains the response values.
Response event: Choose which event of interest the results of the analysis will describe.
Frequency (optional): If the data are in two columns i.e. one column that contains the response values and the other column that contains their frequencies then enter the column that contains the frequencies.
Response in event/trial format: Choose if the response data are two columns – one column that contains the number of successes or events of interest and one column that contains the number of trials.
Event name: Enter a name for the event in the data.
Number of events: Enter the column that contains the number of events.
Number of trials: Enter the column that contains the number of nonevents.
Continuous predictors: Select the continuous variables that explain changes in the response. The predictor is also called the X variable.
Categorical predictors: Select the categorical classifications or group assignments, such as type of raw material, that explain changes in the response. The predictor is also called the X variable.

Step 6: To stores diagnostic measures and characteristics of the estimated equation click Storage… button.

Binary Logistic Regression Storage Dialog Box