Multiple Regression

Multiple regression involves a single dependent variable and two or more independent variables. Multiple regression is basically a statistical technique that simultaneously develops a mathematical relationship between two or more independent variables and an interval scaled dependent variable.

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Questions like how much of the variations in sales can be explained by advertising expenditures, prices and the level of distribution can be answered by employing the statistical technique called multiple regression.

The general form of multiple regression is given by the multiple regression model and is the following:

Y= ß₀ + ß₁X₁ + ß₂X₂ + …….. + ß_kX_k + e.

This multiple regression model is estimated using the following equation:

= a + b₁X₁ + b₂X₂ + …….. + b_kX_k.

There are certain statistics that are used while conducting the analysis on multiple regression.

The R² in multiple regression is the coefficient of the multiple determination. This coefficient in multiple regression measures the strength of association.

The F test in multiple regression is used to test the null hypothesis that the coefficient of the multiple determination in the population is equal to zero.

The partial F test in multiple regression is used to test the significance of a partial regression coefficient. This incremental F statistic in multiple regression is based on the increment in the explained sum of squares that results from the addition of the independent variable to the regression equation after all the independent variables have been included.

The partial regression coefficient in multiple regression is denoted by b₁. This denotes the change in the predicted value per unit change in X_1, when the other independent variables are held constant.

In SPSS, multiple regression is conducted by the researcher by selecting “regression” from the “analyze menu.” From regression, the researcher selects the “linear” option. When the linear regression dialogue box appears, then the researcher enters one numeric dependent variable and two or more independent variables and then finally he will carry out multiple regression in SPSS.

The following assumptions are made in multiple regression statistical analysis:

The first assumption of multiple regression involves the proper specification of the model. This assumption is important in multiple regression because if the relevant variables are omitted from the model, then the common variance which they share with variables that are included in the mode is then wrongly characterized with respect to those variables, and hence the error term is inflated.

The second assumption is that the residual errors in multiple regression are normally distributed. In other words, one can say that the residual errors in multiple regression should follow the normal population having zero as mean and a variance as one.

The third assumption in multiple regression is that of unbounded data. The regression line produced by OLS (ordinary least squares) in multiple regression can be extrapolated in both directions, but is meaningful only within the upper and lower natural bounds of the dependent.

Dissertation Data Analysis

The dissertation is the single most important part of any doctoral student’s career because the dissertation is the last step that a doctoral student must take if he or she is to receive his or her doctoral degree and graduate with the title of “Dr.” The dissertation, then, needs to be completed carefully and meticulously as it will be scrutinized by professors before a student can obtain his or her degree. One of the most difficult aspects of the lengthy and time consuming dissertation is the dissertation data analysis. The dissertation itself revolves around data because the doctoral student must actually prove something new and of interest in the field in which he or she is working. The dissertation data analysis will be that proof and the dissertation data analysis will center the entire dissertation.

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Doctoral students often struggle with the dissertation data analysis because the dissertation data analysis revolves around statistics. In other words, in order to have accurate dissertation data analysis, a doctoral student must be proficient in statistics if he or she is to have valid dissertation data analysis. Dissertation data analysis also takes a lot of time, which is another reason why doctoral students often have a hard time with it. If mistakes are done in the dissertation data analysis phase, then those mistakes from the dissertation data analysis will severely affect the student’s dissertation, conclusion and results. Thus, it is very important that the dissertation data analysis is done in a precise and accurate manner.

Because dissertation data analysis can be so complicated and difficult for doctoral students, there is help available. The first source for help on the dissertation data analysis, of course, is the student’s advisor. The doctoral student’s advisor is not always available to answer questions and provide help on the dissertation data analysis, however, and for that reason, dissertation consultants are the perfect solution to getting help and working on the dissertation data analysis. Dissertation consultants can make sure that the dissertation data analysis is accurate and done correctly because dissertation consultants specialize in dissertation data analysis. The reason why dissertation consultants are a sound solution to acquiring dissertation data analysis help is because dissertation consultants are trained statisticians who specialize in both statistics and in dissertations. Dissertation consultants can look at all of the data that the student has gathered over the course of working on his or her dissertation and dissertation consultants can make sense of that data. Thus, dissertation consultants can be an immense help to students on the dissertation data analysis.

Not only can dissertation consultants help with the dissertation data analysis, however, dissertation consultants can also make sure that the data that has been collected by the doctoral student is valid, accurate and statistically sound. In order to produce accurate dissertation data analysis, one must have valid dissertation statistics and dissertation data. If that data is not valid or accurate, that will of course throw off the dissertation data analysis. Dissertation consultants know this and dissertation consultants will make sure that the data that has been collected has been collected properly, that all of the proper tests were used while collecting the data, that the correct sample sizes were used while collecting the data and that the data is not biased. Once the dissertation consultant has double checked to make sure that the dissertation data is indeed accurate, the dissertation consultants can get to work with the student on the dissertation data analysis. With the help of a dissertation consultant, then, the student will be sure to obtain his or her doctoral degree as every single part of the dissertation data and the dissertation data analysis will be correct, accurate, valid and dependable.

Descriptive Measures

Quantitative data in statistics exhibits some general characteristics that constitute the ideology of descriptive measures.

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There are four different forms of descriptive measures.

The first form of descriptive measures is the measure of central tendency, which is also called the averages.

The second form of descriptive measures is the measure of variation or dispersion.

The third form of descriptive measures is the measure of skewness.

The fourth form of descriptive measures is the measure of kurtosis.

The first form of descriptive measures consists of five descriptive measures, namely Arithmetic Mean, Median, Mode, Geometric Mean and Harmonic Mean.

There are some characteristics that have been put forth by Professor Yule regarding descriptive measures. They are as follows:

1. Descriptive measures should be rigidly defined.

2. Descriptive measures should be less complicated and easy to calculate.

3. The descriptive measures being calculated must be based upon all the observations under consideration.

4. The descriptive measures must be applicable for further mathematical treatment.

5. The descriptive measures must not get affected by the fluctuations of the sampling.

6. The descriptive measures should not get affected by extreme values.

The descriptive measure called the arithmetic mean is defined as the sum of the set of the observations that is divided by the number of that particular set of observations. This descriptive measure satisfies the first five properties laid down by Professor Yule. The biggest disadvantage of this descriptive measure is that it can’t be used in the case of qualitative data and it is also affected by extreme values.

The descriptive measure called the median is defined as that value of the variable which divides the data under consideration into two equal parts. This descriptive measure satisfies the first two and the sixth property put forth by Professor Yule. This descriptive measure can be used in the case of qualitative data, but this descriptive measure cannot be measured quantitatively.

The descriptive measure called mode is defined as the value that occurs most of the time in a particular set of observations. This descriptive measure satisfies the second and the last property that has been put forth by Professor Yule. It is that type of descriptive measure that is used in obtaining an ideal size in business forecasting, etc.

The descriptive measure called the geometric mean is defined as the nth root of the product of the set of the observations under consideration. The basic disadvantage of this type of descriptive measure is that it can neither be easily understood nor be calculated by the person who does not have a mathematical background. This descriptive measure satisfies the first, third, fourth and fifth property put forth by Professor Yule.

The descriptive measure called the harmonic mean is defined as the reciprocal of the arithmetic mean of the reciprocals of the given values provided that none of the observations are zero. The basic disadvantage of this type of descriptive measure is that it cannot be easily understood or be calculated by a person who does not have a mathematical background. This descriptive measure satisfies the first, third, fourth and fifth property put forth by Professor Yule.

The second form of descriptive measure is classified into two categories.

The first category is used in expressing the spread of the observations with respect to the distance that exists between the values of the selected observations. This includes things like range, inter-quartile range, etc.

The second category is used in expressing the spread of the observations with respect to the average of the deviations of the observations for some central value. This includes things like mean, deviation, standard deviation, etc.

The third form of descriptive measure consists of three coefficients of skewness, namely Professor Karl Pearson’s coefficient of skewness, Professor Bowley’s coefficient of skewness and the coefficient of the skewness that is based on the moments.

The fourth form of descriptive measure gives an idea about the flatness or peakedness of the frequency curve. If the curve is neither flat nor peaked, then the descriptive measure concludes that it is a normal curve or a mesokurtic curve. If the curve is flatter than the normal curve, then the descriptive measure concludes that it is a platykurtic curve. If the curve is more peaked, then the descriptive measure concludes that it is a leptokurtic curve.

Binomial Test of Significance

The binomial test of significance is a kind of probability test that is based on various rules of probability. The binomial test of significance is used to examine the distribution of a single dichotomous variable in the case of small samples. The binomial test of significance involves the testing of the difference between a sample proportion and a given proportion.

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This document will discuss certain terms used in the binomial test of significance so that the reader can better understand the binomial test of significance.

The calculation of the binomial test of significance is done in the following manner:

Let us assume that p(r) is to be calculated in the binomial test of significance. In the binomial test of significance, p(r) is the probability that the researcher will obtain an ‘r’ observation in one category of a dichotomy and the researcher will obtain an ‘n – r’ observations in the other category, when the sample size is n. In the binomial test of significance, if ‘p’ is the probability that the researcher will obtain the first category, and ‘q’ is equal to ‘1 – p,’ then it denotes the probability that the researcher will obtain the second category. The formula for the calculation of the binomial test of significance is given by the following:

p(r) = _nC_r*p_r*q_n-r = (n!p_rq_n-r)/(r!(n-r)!)

In this formula of the binomial test of significance, _nC_r is denoted as the number of combinations of n things drawn from ‘r’ at a time.

The normal approximation of the binomial test of significance is made in following manner:

When the size of the sample ‘n’ is greater than 25, and the probability ‘p’ of obtaining the first category is around 0.50, then product of the term ‘npq’ in the binomial test of significance is at least 9. In this case, the binomial distribution approximates the normal distribution in the binomial test of significance. Because of this approximation, a normal curve z-test is used as an approximation of the binomial test of significance. This formula of the approximation of the binomial test of significance is given by the following:

z = ((r[+,-].5) - np)/SQRT(npq)

The binomial test of significance can be done in SPSS. This non parametric test is calculated in SPSS by selecting “Non Parametric test” from the “analyze” menu and then selecting “binomial test of significance.”

There are certain assumptions that are made in the binomial test of significance. The assumptions are the following:

A dichotomous kind of a distribution is assumed in the binomial test of significance. In other words, in the binomial test of significance, it is assumed that the variable of interest is considered to be dichotomous in nature where the two values are mutually exclusive and mutually exhaustive in all cases being considered. The word ‘binomial’ in the binomial test of significance suggests that the variables of interest should be dichotomous in nature as the term ‘binomial’ means two.

Since this binomial test of significance does not involve any parameter and therefore is non parametric in nature, the assumption that is made about the distribution in the parametric test is therefore not assumed in the binomial test of significance.

In the binomial test of significance, it is assumed that the sample that has been drawn from some population is done by the process of random sampling. The sample on which the binomial test of significance is conducted by the researcher is therefore a random sample.