The Assumption of Homogeneity of Variance

The assumption of homogeneity of variance is an assumption of the ANOVA that assumes that all groups have the same or similar variance.  The ANOVA utilizes the F statistic, which is robust to the assumption, as long as group sizes are equal.  Equal group sizes may be defined by the ratio of the largest to smallest group being less than 1.5.  If group sizes are vastly unequal and homogeneity of variance is violated, then the F statistic is considered liberal when large sample variances are associated with small group sizes.  When this occurs, the alpha value is greater than the level of significance.  This indicates that the null hypothesis is being falsely rejected.  On the other hand, the F statistic is considered too conservative if large variances are associated with large group sizes.  This would mean that the actual alpha value is less than the level of significance.  This does not cause the same problems as falsely rejecting the null hypothesis, however, it can cause a decrease in the power of the study.  

To test for homogeneity of variance there are several statistical tests that can be used; these tests include: Hartley’s F_max, Cochran’s, Levene’s and Barlett’s test.  Several of these assessments have been found to be too sensitive to non-normality and are not frequently used.  Of these tests, a more common assessment for homogeneity of variance is Levene’s test.  The test statistic for Levene’s test is calculated by diverging the data for each group from the group mean, and then comparing the absolute values.  Levene’s test is presented with the F statistic, as an ANOVA is conducted to compare the absolute values.  A p value less than .05 indicates a violation of the assumption.  If a violation occurs, it is likely that conducting the non-parametric equivalent of the analysis is more appropriate. 

ANOVAs and MANOVAs

An ANOVAs is used to assess differences on time and/or group for one continuous variable and a MANOVA is used to assess differences on time and/or group for multiple continuous variables, but what other factors go into the decision to conduct multiple ANOVAs or a single MANOVA?  MANOVAs are best conducted when the dependent variables used in the analysis are highly negatively correlated and are also acceptable if the dependent variables are found to be correlated around .60, either positive or negative.  The use of MANOVA is discouraged when the dependent variables are not related or highly positively correlated.  

MANOVA is discouraged with highly positively correlated variables because, although the overall multivariate analysis works well, once the highest priority dependent variables has been assessed, the tests conducted and results presented on the remaining dependent variables will be vague.  The reason for this is because once the highest priority dependent variable becomes a covariate, the variance that remains for the lower priority dependent variables is not enough to be significantly related to the main effects or interactions.  Additionally, the univariate ANOVA results are misleading.  

MANOVA is also discouraged when the dependent variables are not significantly related.  A multivariate analysis has lower power than univariate analyses, therefore the difference between univariate and step-down analysis is small.  In this instance the only benefit to conducting a MANOVA over univariate ANOVAs is a reduction in the likelihood of Type I error.  If multiple ANOVAs are the more appropriate analysis, Type I error can be controlled for with the use of the Bonferroni correction, α = 1 - (1 - α₁)(1 - α₂)…(1 - α_n).

In the case where some the dependent variables are correlated in different sets, it may be more appropriate to run two separate MANVOAs; one with each set of correlated variables.  

Independent and Dependent Variables

To understand the concept of independent and dependent variables, one should understand the meaning of variables. Variables are defined as the properties or kinds of characteristics of certain events or objects.

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Independent variables are variables that are manipulated or are changed by researchers and whose effects are measured and compared. The other name for independent variables is Predictor(s). The independent variables are called as such because independent variables predict or forecast the values of the dependent variable in the model.

The other variable(s) are also considered the dependent variable(s). The dependent variables refer to that type of variable that measures the affect of the independent variable(s) on the test units. We can also say that the dependent variables are the types of variables that are completely dependent on the independent variable(s). The other name for the dependent variable is the Predicted variable(s). The dependent variables are named as such because they are the values that are predicted or assumed by the predictor / independent variables. For example, a student’s score could be a dependent variable because it could change depending on several factors, such as how much he studied, how much sleep he got the night before he took the test, or even how hungry he was when he took it. Usually when one is looking for a relationship between two things, one is trying to find out what makes the dependent variable change the way it does.

Let us identify independent and dependent variables in the following cases:
In the case of a linear model, we have the general equation as:

Here, Y is the variable dependent on X, therefore, X, is an independent variable.

Similarly, in cases of the regression model, we have

Here, the regressors, ßij (j=1, p) are the independent variables and the regressands Yi are the dependent variables.

Independent variables are also called “regressors,“ “controlled variable,” “manipulated variable,” “explanatory variable,” “exposure variable,” and/or “input variable.” Similarly, dependent variables are also called "response variable," "regressand," "measured variable," "observed variable," "responding variable," "explained variable," "outcome variable," "experimental variable," and/or "output variable."

A few examples can highlight the importance and usage of dependent and independent variables in a broader sense.

If one wants to measure the influence of different quantities of nutrient intake on the growth of an infant, then the amount of nutrient intake can be the independent variable, with the dependent variable as the growth of an infant measured by height, weight or other factor(s) as per the requirements of the experiment.

If one wants to estimate the cost of living of an individual, then the factors such as salary, age, marital status, etc. are independent variables, while the cost of living of a person is highly dependent on such factors. Therefore, they are designated as the dependent variable.

In the case of time series analysis, forecasting a price value of a particular commodity is again dependent on various factors as per the study. Suppose we want to forecast the value of gold, for example. In this case the seasonal factor can be an independent variable on which the price value of gold will depend.

In the case of a poor performance of a student in an examination, the independent variables can be the factors like the student not attending classes regularly, poor memory, etc., and these will reflect the grade of the student. Here, the dependent variable is the test score of the student.

Statistics Consulting Services

The global economy pushes itself to development, with ever increasing competency and competition as access to information becomes more readily available. Necessity has required statistics consulting services, and these statistics consulting services offer professional statisticians who can specifically answer the needs of emergent markets. The existence of the different fields requiring statistics has created the need for these statistics consulting firms. These different fields include sciences like sociology, psychology, education, government and law, commerce and medicine.

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Statistics consulting services require professionals who are extremely competent because intellectual research and analysis requires discipline and training. Clients who seek the advantage of statistics consulting services are at an advantage because they get dependable data analysis. The following are some advantages of signing up for statistics consulting services:


1. Clients are exposed to a wide variety of services when investing in statistics consulting services, as the researchers have considerable experience in research and statistics.

2. There are built-in designs and dependable systems in applying statistical analysis and they are tested and tried by statistics consulting services.

3. On-time reports and interpretations of data are given to clients through statistics consulting services.

4. Statistics consulting services give quality results as they execute the job and get the right statistics tools needed for the study.

5. Professionals who are involved with statistics consulting services are competent and dependable, as they are trained to work rigorously as they give accurate interpretation of figures and facts.

6. The ease of contacting statistics consulting services saves time. Additionally, the availability of personnel will guarantee the client’s satisfaction.

7. Statistics consulting services are staffed by professional statisticians.

8. The price is reasonable in statistics consulting services because the value of money and time is multiplied as professional partnerships are forged.

10. Statistics consulting services assure the clients get assistance, advice and support throughout the project.



Theses, reports and market studies need statistics consulting services to make their research viable and dependable. When choosing a statistics consulting services, it is best to check on the track record of the service, and the track record of the professionals working with the consulting firm. It is important to review the reputation of the institution that claims to provide the statistics consulting services. Before finalizing a project with the statistics consulting services, the need to collaborate must be clear to both parties.

Consultants from statistics consulting services have good communication skills and build professional interpersonal relationships. Writing theses, market studies and data analysis, can be difficult, but it is made much easier with statistics consulting services.

Organizational problems are minimized and time is managed when statistics consulting services are appointed—especially for fast track trading companies and busy student-young-professionals, who are struggling to finish their theses and dissertations. Through statistics consulting services, clients are able to maximize their time.

Students and young professionals, whose niche is different than statistical analysis, are greatly benefited because they discover the advantages of a collaborative learning atmosphere with statistics consulting services. Most people who are finishing up their career courses who opt to invest in statistics consulting services are better prepared to finish ahead of schedule. They can then move towards advancement, as others struggle to finish their work because they are without the help of statistics consulting services.

It is important for students to finish up their theses or dissertation, thus they need statistics consulting services. More often than not, students are experts in their specific fields, but the need to know the nitty-gritty of statistics makes them in need of statistics consulting services.

Statistics consulting services assist the students in verifying the certainty of facts of the data gathered and therefore it strengthens the development of the theses. Students are helped in editing and systematizing the collection of data. As our global economy moves ahead quickly, it is important to strategically plan to efficiently deliver studies accurately and on time. Statistics consulting services makes this possible.

Discriminant Analysis

During a study, there are often questions that strike the researcher that must be answered. These questions include questions like ‘are the groups different?’, ‘on what variables, are the groups most different?’, ‘can one predict which group a person belongs to using such variables?’ etc. In answering such questions, discriminant analysis is quiet helpful.

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Discriminant analysis is a technique that is used by the researcher to analyze the research data when the criterion or the dependent variable is categorical and the predictor or the independent variable is interval in nature. The term categorical variable in discriminant analysis means that the dependent variable is divided into a number of categories. For example, three brands of computers, Computer A, Computer B and Computer C can be the categorical dependent variable in Discriminant Analysis.

The objective of discriminant analysis is to develop discriminant functions that are nothing but the linear combination of independent variables that will discriminate between the categories of the dependent variable in a perfect manner. Discriminant analysis enables the researcher to examine whether significant differences exist among the groups, in terms of the predictor variables. Discriminant analysis evaluates the accuracy of the classification.

Discriminant analysis is described by the number of categories that is possessed by the dependent variable.

As in statistics, everything is assumed up until infinity, so in this case, when the dependent variable has two categories, then the type of discriminant analysis used is two-group discriminant analysis. If the dependent variable has three or more than three categories, then the type of discriminant analysis used is multiple discriminant analysis. The major distinction to the types of discriminant analysis is that for a two group discriminant analysis, it is possible to derive only one discriminant function. On the other hand, in the case of multiple discriminant analysis, more than one discriminant function can be computed.

There are many examples that can explain when discriminant analysis fits. Discriminant analysis can be used to know whether heavy, medium and light users of soft drinks are different in terms of their consumption of frozen foods. In the field of psychology, discriminant analysis can be used to differentiate between the price sensitive and non price sensitive buyers of groceries in terms of their psychological attributes or characteristics. In the field of business, discriminant analysis can be used to understand the characteristics or the attributes of a customer possessing store loyalty and a customer who does not have store loyalty.

For a researcher, it is important to understand the relationship of discriminant analysis with Regression and Analysis of Variance (ANOVA) which has many similarities and differences. Often we can find similarities and differences with the people we come across. Similarly, there are some similarities and differences with discriminant analysis along with two other procedures. The similarity is that the number of dependent variables is one in discriminant analysis and in the other two procedures, the number of independent variables are multiple in discriminant analysis. The difference is categorical or binary in discriminant analysis, but metric in the other two procedures. The nature of the independent variables is categorical in Analysis of Variance (ANOVA), but metric in regression and discriminant analysis.

The steps involved in conducting discriminant analysis are as follows:
• The problem is formulated before conducting the discriminant analysis.
• The discriminant function coefficients are estimated in discriminant analysis.
• The next step is the determination of the significance of these discriminant functions in discriminant analysis.
• One must interpret the results obtained in discriminant analysis.
• The last and the most important step is to assess the validity of discriminant analysis.

Statistics Consultant

To their disadvantage, many dissertation writing students do not know that statistics consultants are available and ready to help a dissertation writing student through the very lengthy and challenging aspects of the dissertation. In fact, a statistics consultant can mean the difference between a doctoral student struggling through the entirety of the dissertation and a doctoral student finishing the dissertation on-time and with great success.

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One of the most challenging aspects of any dissertation is the statistical procedures that accompany the dissertation. The dissertation statistics are long and complicated because they must validate whatever it is the doctoral seeking student is proving—and this is made even more complicated and difficult if the student is new to the dissertation writing process. Students who are writing a dissertation for the very first time struggle mightily when it comes to the dissertation statistics because students who have never written a dissertation before have simply not been exposed to the complicated statistical procedures involved in each and every single dissertation.

A statistics consultant can help these “first timers,” however, as a statistics consultant can take the dissertation writing student through the dissertation statistics step by step. Thus, a statistics consultant can help the student from the very first stages of the dissertation and a statistics consultant will stay on until the very end of the project.

More specifically, a statistics consultant can help a dissertation writing student by helping the student with the following aspects of the dissertation:
• A statistics consultant can help the student narrow down a topic that is manageable and able to be studied statistically.
• A statistics consultant can help a student phrase that topic so that it uses proper “statistical vocabulary.”
• A statistics consultant can help the student perform the extensive dissertation research that needs to be performed before starting the dissertation statistics.
• A statistics consultant can help the student with the proposal phase of the dissertation
• A statistics consultant can help the student with the dissertation methodology that needs to be completed
• A statistics consultant can help the doctoral student as he or she gathers the data that is necessary for the statistics. Thus a statistics consultant can help ensure that the data that is gathered is appropriate, is not biased and is of the right sample size
• A statistics consultant can help the doctoral student make sense of the numbers and data that has been gathered. Thus a statistics consultant can help analyze, collate and interpret the data
• A statistics consultant can ensure that the dissertation writing student has applied the statistics properly to the dissertation
• A statistics consultant can help the student with the ‘finishing touches’ of the dissertation—so a statistics consultant will proofread the entire dissertation and a statistics consultant will make sure that the dissertation is ready to be turned in and that it will be both accepted and approved the first time that it is submitted by the doctoral student.

One of the most important aspects about hiring a statistics consultant to help you with your dissertation is the fact that by using a statistics consultant, you will be ensuring that you finish the dissertation on time and with success. Even more crucial than that, however, is the fact that the statistics consultant will take the time to explain everything to you and a statistics consultant will actually instruct and teach you everything you need to know about statistics and statistical procedures. And this instruction will prove infinitely useful and constructive in the future.

Statistics Consulting

Statistics consulting has been sought by thousands of doctoral degree seeking students as they write the lengthy and challenging dissertation. Statistics consulting has been sought by these students because statistics consulting is the best way to get help on the difficult statistical procedures that accompany the dissertation. This document will detail the benefits of statistics consulting.

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Statistics consulting begins when the dissertation writing student contacts a dissertation consulting firm. A dissertation consulting firm is staffed with expert statisticians who are trained in both the dissertation and in statistics—both of which go hand in hand on every single dissertation. Statistics consulting, then, begins the moment a student decides to contact a dissertation consulting firm, and because this is true, the sooner the student gets the help that he or she needs, the more the statistics consulting can benefit the student.

Once the student decides that it is not worth struggling alone, and that statistics consulting is the best way to get help, the student will be paired up with a dissertation statistics consultant and the statistics consulting will begin. The dissertation statistics consultant will go over everything that the student needs to do in order to obtain accurate and precise results. Thus, the dissertation statistics consultant will go through everything that has already been done and go through everything that has yet to be done.

Most often, the dissertation writing student is struggling on the statistical aspects of the dissertation. This is quite common because the statistical procedures of the dissertation are the most challenging aspect of the dissertation. Not so, however, with statistics consulting. Statistics consulting will provide the student with every single thing that the student needs to conquer the complicated dissertation statistics. Statistics consulting, then, provides help on the gathering of the data; statistics consulting provides help on making sense of the data; statistics consulting provides help on translating the data into charts, graphs and figures; statistics consulting helps the student as the student must apply the statistics to the dissertation; and statistics consulting helps the student on the final phases of the dissertation—on the proofreading of the dissertation.

Statistics consulting, then, will go through every single step of the statistics with the dissertation writing student and statistics consulting will not allow the student to make any mistakes when it comes to the dissertation and the statistical portions of the dissertation.

Perhaps the best service that statistics consulting provides, however, is instruction. It is very important for the PhD candidate to actually understand every single statistical procedure that is contained within his or her dissertation. This is important for many reasons. The most important reason, however, is because the PhD candidate will have to orally defend his or her dissertation. In this oral defense, the dissertation writing student will be “grilled” (or questioned heavily and extensively) on the dissertation statistics. If the student is unable to answer those questions accurately, he or she will not get the PhD that he or she has been striving for. Thus, statistics consulting will make sure that each and every single student who seeks statistics consulting actually understands every single part of the statistics that is contained within the dissertation. With the help of statistics consulting, then, the student will be well on his or her way to receiving his or her doctoral degree while actually understanding the difficult concepts of the statistical procedures.