Statistical Tests

Statistical tests are of various types, depending upon the nature of the study. Statistical tests provide a method for making quantitative decisions about a particular sample. Statistical tests mainly test the hypothesis that is made about the significance of an observed sample.

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There are some key concepts of statistical tests that can help in understanding statistical tests.

Type I error: Type I error in a statistical test is usually committed when a correct sample is rejected.

Type II error: Type II error in a statistical test is usually committed when a false sample is accepted.

We can say that statistical tests are generally categorized into various types depending upon the type of field. Statistical tests are carried out extensively in psychology, medicine, nursing and business.

In the field of psychology, statistical tests of significances like t-test, z test, f test, chi square test, etc., are carried out to test the significance between the observed samples and the hypothetical or expected samples. For example, if a researcher wants to conduct a statistical test upon the significant difference between the IQ levels of two college students, then the researcher can perform the t statistical test for the difference of the two samples. If one wants to test the goodness of fit of a particular assumed model, then one can use the chi square test of goodness of fit. This is the only statistical test (among other statistical tests), that helps in testing the goodness of fit of an assumed model.

In the field of business, statistical tests are used for conducting Analysis of variance (ANOVA). Analysis of variance (ANOVA) is basically used for examining the differences in the mean values of the dependent variables associated with the effect of the controlled independent variables, after taking into account the influence of the uncontrolled independent variables. Thus, in Analysis of variance (ANOVA), f statistical test calculates the significance of the samples.

Statistical tests are directly correlated to statistical inference. Statistical inference involves tests of hypothesis, where statistical tests play a crucial role. In the field of medicine and nursing, tests of hypotheses are conducted using various statistical tests. If the researcher makes mistakes in calculation while performing the statistical tests, then the researcher might end up committing a type II error. In other words, if the researcher makes a mistake in calculation, then the statistical tests will conclude that a false drug sample is a correct drug sample. Further, the researcher might end up tagging a false drug sample as a correct drug sample. Thus, the researcher should be cautious while performing statistical tests. In the field of medicine and nursing, errors in statistical tests can result in huge problems in people’s lives, as it affects their drugs and dosages etc.

Statistical tests can be performed in software such as SPSS. Statistical tests in SPSS can be performed with the help of the “analysis” menu. For every statistical test, there are different sample sizes. In the t test, for example, we should have a sample size less than 30. Similarly, for a statistical z test, we should have a sample size more than 30. Statistical tests come with some general assumptions like the assumption that samples should be drawn from the population in a random manner. The observations in statistical tests must be independent. The observations in statistical tests must have the same underlying distribution. Especially, in the chi sq statistical test, observations must be grouped in different categories. Normal distribution of deviations is assumed in statistical tests.

Monte Carlo Methods

Monte Carlo methods are methods that help the researcher in estimating solutions. This in turn helps the researcher in addressing a variety of mathematical problems that involve several statistical sampling experiments. The Monte Carlo methods are nothing but the collection of the different types of procedures that perform the same operations. The Monte Carlo methods iteratively evaluate the deterministic model by utilizing the method of random numbers and the theory of probability for getting an almost-accurate answer for the problem.

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The researchers should note that Monte Carlo methods only help in obtaining an approximate outcome. Therefore, when a researcher uses the Monte Carlo methods, the error approximation is the crucial factor. While using the Monte Carlo methods, the researcher must take the error approximation factor into account.

The various types of Monte Carlo methods used by the researcher have relatively different levels of exactness. This also depends upon the type of question or problem that is to be addressed by the investigator. The Monte Carlo methods are applicable because they consist of the computation of difficult integrals. The Monte Carlo methods are used in those cases where multi dimensional integrals are involved. The Monte Carlo methods are useful in those cases where logical approximation is required.

There are several Monte Carlo methods.

A method called the crude Monte Carlo method is a type of method that is used to solve the integral of a particular function, f(x), for example if it falls under the limits ‘a’ and ‘b.’ In this Monte Carlo method, the researcher selects a number ‘N’ from the random sample, ‘s.’ In this Monte Carlo method, the limit ‘a’ and ‘b’ on which the function is integrated is not equal to the value of the sample size. After this, the investigator then locates the function value f(s) in the function f(x) for each random sample‘s’ in the Monte Carlo method. After locating, the researcher then performs the addition of all these values and divides the sum by ‘N,’ which gives the mean values from the sample in the Monte Carlo methods.

The investigator then multiplies the value in order to obtain the integral in the Monte Carlo methods.

A method called the acceptance rejection Monte Carlo method is that type of Monte Carlo method that is a flexible technique and is simple to understand. However, this type of Monte Carlo method provides the researcher with one of the least approximate results among all four types of Monte Carlo methods.

The reason behind using this type of Monte Carlo method is that it is useful for the researcher to obtain the variance by simply adding up the variances for each sub interval.

The first two Monte Carlo methods that are discussed are generally the two basic important techniques that are important for the researcher to understand. As such, Monte Carlo methods are used extensively since they serve as a basis in more complex techniques.

The Monte Carlo methods are widely used in various disciplines like physics and chemistry, as they simulate the tedious reactions and interactions.

There is also a smoothening property in the Monte Carlo methods that is useful in the case of complex problems. Approximation of the complex problems is generally very time-consuming but the Monte Carlo methods make it easy.

The Monte Carlo methods can be used in the field of computer vision. The Monte Carlo methods in the field of computer vision are used for object tracking.

Path Analysis

Path analysis represents an attempt to deal with causal types of relationships. Path analysis was developed by Sewall Wright in 1930. Path analysis is very useful in illustrating the number of issues that are involved in causal analysis.

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Path Analysis can be carried out by the researcher diagrammatically or graphically in the form of circles and arrows, which indicate various causations among variables. The ultimate goal of path analysis is to predict the regression weight. The regression weight is predicted during path analysis, and then compared to the observed correlation matrix. Path analysis is also applicable in cases where the researcher wants to perform a goodness of fit test.

There are certain terminologies that are used in Path analysis.

The exogenous variables in Path analysis are variables whose causes are outside of the model.
The endogenous variables in Path are variables whose causes are inside the model.

The recursive model in Path analysis is a causal model that is unidirectional. In other words, they have one way causal flow. This model in Path analysis has neither feedback loops nor any reciprocal effects. In this type of model in Path analysis, the variables cannot be both cause and affect at the same time.

The non recursive model in Path analysis is a causal model with feedback loops and reciprocal effects.

The path coefficient in Path analysis is the standardized regression coefficient that predicts one variable from another.

Path analysis is usually conducted with the help of an added module called the analysis of moment structures (AMOS). Other than the added module of SPSS called the analysis of moment structures (AMOS), there is other statistical software like SAS, LISREL, etc. that can be used to conduct path analysis. According to a well known researcher named Kline (1998), an adequate sample size should always be 10 times the amount of the parameters in path analysis.

The best sample size should be 20 times the number of parameters in path analysis.

Since path analysis is also a kind of statistical analysis, path analysis also comes with several assumptions.

In path analysis, the association among the model should be linear in nature. In path analysis, the associations among the models should be additive in nature. In path analysis, the association among the model should be causal in nature. The data that is used in path analysis should follow an interval type of scale. In order to reduce volatilities in the data, it is assumed in the theory of path analysis that all the error terms in path analysis are not correlated among the various variables. In path analysis, is it is also assumed that errors are not correlated among themselves. In path analysis, it is assumed that there is only one way causal flow.

Path analysis does have some limitations.

Path analysis can very well evaluate, test or compute two or more than two types of causal hypotheses. However, the major limitation of path analysis is that path analysis cannot establish the direction of causality.

Path analysis is applicable only in those kinds of cases where relatively small numbers of hypotheses can be easily represented by a single path.

Multicollinearity

The term multicollinearity was first used by Ragnar Frisch. Multicollinearity means that there is a perfect or exact relationship between the regression exploratory variables. Linear regression analysis assumes that there is no perfect exact relationship among exploratory variables. In regression analysis, when this assumption is violated, the problem of Multicollinearity occurs.

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In regression analysis, multicollinearity has the following types:

1. No multicollinearity: When the regression exploratory variables have no relationship with each other, then there is no multicollinearity in the data.
2. Low multicollinearity: When there is a relationship among the exploratory variables, but it is very low, then it is a type of low multicollinearity.
3. Moderate multicollinearity: When the relationship among the exploratory variables is moderate, then it is said to be moderate multicollinearity.
4. High multicollinearity: When the relationship among the exploratory variables is high or there is perfect correlation among them, then it said to be high multicollinearity.
5. Very high multicollinearity: When the relationship among the exploratory variables is exact, then it is the problem of very high multicollinearity, which should be removed from the data when regression analysis is conducted.

Many Factors affect multicollinearity. For example, multicollinearity may exist during the data collection process, or multicollinearity may exist due to the wrong selection of the model. For example, if we take the exploratory variables to be income and house size in our model, then the model will have the problem of multicollinearity because income and house size are highly correlated. Multicollinearity may also occur if we take too many exploratory variables in regression analysis.

Consequences of multicollinearity: If the data has a perfect or exact multicollinearity problem, then the following will be the impact of multicollinearity:

1. In the presence of multicollinearity, variance and covariance will be wider, which will make it difficult to reach a statistical decision for the null and alternative hypothesis.
2. In the presence of multicollinearity, the confidence interval will be wider due to the wider confidence interval. In this case, we will accept the null hypothesis, which should be rejected.
3. In the presence of multicollinearity, the standard error will increase and it makes the value of the t-test smaller. We will accept the null hypothesis that should be rejected.
4. Multicollinearity will increase the R-square as well, which will impact the goodness of fit of the model.

Detection of multicollinearity: The following are the methods that show the presence of multicollinearity:

1. In regression analysis, when R-square of the model is very high but there are very few significant t ratios, this shows multicollinearity in the data.
2. High correlation between exploratory variables also indicates the problem of multicollinearity.
3. Tolerance limit and variance inflating factor: In regression analysis, one-by-one minus correlation of the exploratory variable is called the variance inflating factor. As the correlation between the repressor variable increases, VIF also increases. More VIF shows the presence of multicollinearity. The inverse of VIF is called Tolerance. So the VIF and TOI have a direct connection.

Remedial measure of multicollinearity: In regression analysis, the first step is to detect multicollinearity. If multicollinearity is present in the data, then we can solve this problem by taking several steps. The first step is to drop the variable, which has the specification bias of multicollinearity. By combining the cross sectional data and the time series data, multicollinearity can be removed. If there is a high multicollinearity, then it can be removed by transforming the variable. By taking the first or the second, different variables can be transformed. By adding some new data, multicollinearity can be removed. In multivariate analysis, by taking the common score of the multicollinearity variable, multicollinearity can be removed. In factor analysis, principle component analysis is used to drive the common score of multicollinearity variables. A rule of thumb to detect multicollinearity is that when the VIF is greater than 10, then there is a problem of multicollinearity.

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Dissertation Statistics Consultants

Dissertation statistics consultants are the most efficient and affordable way to get help on your dissertation. Dissertation statistics consultants are consultants who are experts at both dissertations and at statistics, and as such, dissertation statistics consultants can help any student with his or her dissertation needs.

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Dissertation statistics consultants are experts at both dissertations and statistics because dissertation statistics consultants are trained statisticians. Additionally, dissertation statistics consultants have provided help to thousands of students who need to finish a dissertation, and thus, they are also extremely well trained when it comes to working on a dissertation and ensuring that it is approved. In other words, the statistical skills of a dissertation statistics consultant combined with the experience of helping students with dissertations is the perfect combination of skills and expertise need to help any student finish his or her dissertation.

The dissertation itself is a very lengthy and difficult process, and chances are that if you have started this long and difficult project, that you know just how tedious and time consuming it is. In fact, many students often consider it to be the most challenging aspect of their entire student career. Add to all of this the pressure and stress that go along with having to finish it on time and successfully, and students simply become overwhelmed by the entire process of the dissertation. Dissertation statistics consultants can help these students, however, as dissertation statistics consultants can offer invaluable guidance, support and assistance throughout the entire dissertation process. And there is no better way to alleviate stress and assuage fears than to get help and assistance. Thus, dissertation statistics consultants can be the most invaluable tool when it comes to finishing a dissertation on-time and with success.

Dissertation statistics consultants will make the entire process of writing and working on your dissertation more efficient. Dissertation statistics consultants will do this by making sure that a student is taking every single correct step when it comes to the dissertation and the methodology that the student is using to complete his or her dissertation. In other words, the dissertation statistics consultant will ‘break everything down’ into little steps that are manageable, understandable and doable, and this will make the process of completing a dissertation much easier. Additionally, with the help of dissertation statistics consultants, the student will not make any missteps. A misstep occurs when a student somehow makes a mistake in terms of the methodology of a dissertation. These missteps are quite common and occur in the statistical part of the dissertation. Because statistics is a science and requires students to follow procedures, rules and guidelines very precisely, there is much room for students to make mistakes. And when a mistake or misstep occurs, the student must start that section over again. And this is incredibly time consuming. Dissertation statistics consultants will not let this happen, however, as dissertation statistics consultants are trained professionals when it comes to statistics. Additionally, they know the common missteps that are likely to happen—for example, using the wrong sample size when gathering data, or using the wrong test to interpret the data that has been gathered, etc—and dissertation statistics consultants will make sure that the student does not waste time and energy by falling into the common pitfalls of a dissertation. Thus, with the help of dissertation statistics consultants, the student will be able to finish his or her dissertation on time because he or she does not need to waste any time making very costly mistakes.

There is no doubt that dissertation statistics consultants are the best way to ensure results. Whether you are at the beginning, the middle or the end of your project, you should look into hiring dissertation statistics consultants as they can guarantee that you finish your dissertation successfully, on-time, and with confidence.

Dissertation Statistics Consultant

While it is extremely exciting to get to the point where a student is finally ready to begin his or her dissertation, that excitement is often short-lived as the student realizes exactly what needs to be done in order to successfully complete his or her dissertation. Because the dissertation is necessary to attain a doctoral degree, that excitement can quickly turn to stress, nervousness, anxiety and fear.

There is no need to stress over a dissertation, however, as a dissertation statistics consultant can help any student that needs to finish a dissertation. Statistics Solutions, Inc. is the country's leader in dissertation statistics consulting. Contact Statistics Solutions today for a free 30-minute consultation on your dissertation statistics.

A dissertation statistics consultant is a trained professional who offers help, assistance and guidance to a student throughout the dissertation writing process. With the help of a dissertation statistics consultant, the student will quickly learn that it is not necessary to worry over the completion of his or her dissertation as a dissertation statistics consultant can ease the pressures and worries of a student.

Because the dissertation statistics consultant is trained to help students, and because the dissertation statistics consultant has helped many students in the past, the dissertation statistics consultant is very well aware of the pressure and anxiety that students feel as they work on their dissertation. For this reason, one of the first things that a dissertation statistics consultant does is to come up with a plan of action. This plan of action includes building a timeline with the student so the student knows exactly what needs to be done and when it needs to be done. This timeline that is created with the dissertation statistics consultant will detail specific targets and due-dates. Thus, with the timeline created by the dissertation statistics consultant, the student can see exactly what needs to be done. This is one sure way to alleviate the stress and pressures that students who work on their dissertation feel. This is because building a timeline lessens the ‘mystery’ of what needs to be done next.

Not only will the dissertation statistics consultant build a reliable and easy-to-follow timeline, the dissertation statistics consultant will also tell the student exactly how to do everything on that timeline. Thus, the student who has the help of the dissertation statistics consultant will have the tools necessary to carry out the procedures and methodology of his or her dissertation.

One of the most time consuming aspects of the dissertation is the obtaining of valid and useful statistics. The statistics will eventually prove the dissertation, or a student’s point, so it is incredibly important to get accurate and valid statistics. With the help of a dissertation statistics consultant, this too is easy and manageable. Because a dissertation statistics consultant is a trained and expert statistician, the dissertation statistics consultant knows how to perform statistical analysis. And while everything in statistical analysis can be time-consuming and difficult, this is not true with the help of a dissertation statistics consultant. Further the dissertation statistics consultant will explain every single aspect of the statistics. This is incredibly helpful because most doctoral students are not adequately trained in statistics. While they are thoroughly trained and knowledgeable in their field of study, they are not trained properly in statistics. Thus, one of the most beneficial things that a dissertation statistics consultant provides is instruction. This instruction, provided by the dissertation statistics consultant, goes a long way in ensuring that the student succeeds with his or her dissertation. This is true because the student must orally defend his or her dissertation, and it is important for the student to have a working knowledge of what has been completed statistically in his or her dissertation. The dissertation statistics consultant will provide that working knowledge and the dissertation statistics consultant will guarantee that a student finishes his or her dissertation with success.

Continuous Probability Distribution

The continuous probability distribution is basically a kind of distribution that is based on the continuous type of random variables. The continuous type of random variable deals with the continuous probability distribution.

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The continuous type of variable in continuous probability distribution consists of the probability density function, called the pdf. Because in the continuous probability distribution the variable is not countable, it is measured with respect to the density.

A normal distribution in the continuous probability distribution generally falls in between the range of -∞ to +∞. This continuous probability distribution has the parameter as µ (called the mean) and σ2 (called the variance). The probability density function (pdf) of this continuous probability distribution is being given by the following:
f(x;µ, σ)= (1/ σ π) exp(-0.5 (x-µ)2/ σ2).

This kind of continuous probability distribution has an important role in statistical theory for several reasons.

The distributions, like the Binomial distribution, Poisson distribution and Hyper Geometric distribution, are approximated by using the continuous probability distribution.

This type of continuous probability distribution is very much applicable in the Statistical Quality Control (SQC).

This type of continuous probability distribution is used in the study of large sample theory, when the normality is involved. The study of the sample statistics is done with the help of the curves of this type of continuous probability distribution.

The theory on which the significance tests (like the t-test or the F test) are based use assumptions that are assumed on the parent population, which belongs to this type of continuous probability distribution.

A continuous probability distribution called the gamma distribution generally falls in the range of 0 to ∞. This type of continuous probability distribution has the parameter ‘d>0’. The probability density function (pdf) of the continuous probability distribution is given by the following:

f(x)= exp(-x) xd-1/

This continuous probability distribution has a property called the additive property, which tells that the sum of the independent variables of this continuous probability distribution is equal to the variable of this continuous probability distribution.

A continuous probability distribution called beta distribution of the first kind has the range which ranges between 0 and 1. The parameters of this type of continuous probability distribution is µ>0 and v>0. The probability density function (pdf) of this type of continuous probability distribution is given by the following:

f(x)= (1/B(µ,v)) xµ-1 (1-x)v-1

A continuous probability distribution called the beta distribution of the second kind falls in the range of 0 to ∞. This type of continuous probability distribution has the parameters namely µ>0 and v>0.The probability density function (pdf) of this continuous probability distribution is given by the following:

f(x)= (1/B(µ,v)) xµ-1 (1+x)v+µ

The continuous probability distribution called the Weibul distribution falls in the range of µ to ∞. This type of continuous probability distribution consists of three parameters, namely c(>0), α(>0) and µ. The probability density function (pdf) in continuous probability distribution is given by the following:

f(x;c,α,µ) = (c (x-µ/α)c-1)/ α exp (-(x-µ/α)c)

A logistic distribution is a continuous probability distribution with the parameter α and β. This type of continuous probability distribution is used widely as a growth function in population and other demographic studies. This type of continuous probability distribution is considered as the mixture of the extreme values of the distributions.

A Cauchy distribution is a continuous probability distribution with the parameter ‘l’ > 0 and ‘µ’. This type of continuous probability distribution has the range of -∞ to +∞. The continuous probability distribution is given by the following:

f(x)= l/π(l2+(x-µ)2)

Statistical Consultants

If you are struggling with statistics, you are most definitely wasting valuable time! This is because there are trained statisticians who can help you with all of your statistical needs. What’s more, because these statistical consultants are trained statisticians, they can perform statistics efficiently and accurately. Thus, there is no need to struggle with statistics when it is cost effective and relatively inexpensive to hire statistical consultants.

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Statistical consultants are available to help anyone who needs statistical help. And because statistics has become increasingly important in today’s world, the demand for statistical consultants continues to increase. The reason for this is simple: proper statistics increases productivity, efficiency, sales and profit. Any business, then, can benefit from the use of statistics and from the people who can accurately obtain those statistics. Statistical consultants are the perfect fit for businesses as statistical consultants can be hired to analyze the efficiency of a business. Statistical consultants can see what does and does not work in terms of product price, product placement, product location, etc. Additionally, statistical consultants can provide statistics on every single aspect of business.

Statistical consultants can be especially beneficial to small businesses as small businesses often do not have people who are trained in statistics. Instead, small businesses rely on intuition, something that can be very costly for a business owner. Instead of relying on intuition, however, small businesses should seek the help of statistical consultants to provide them with the information that they need to maximize profit.

Just at statistical consultants can help small business owners, statistical consultants can be helpful to students who need to use statistics to write a dissertation. Because not all students are well versed in statistics, statistical consultants can step-in and fill in the gaps in terms of what needs to be done to produce accurate statistics. Statistical consultants, then, can gather the data necessary on which to base all of the dissertation statistics. This data collecting must be done very methodically and precisely, otherwise the data will be skewed and the entire dissertation will be void, for it is impossible to have an accurate dissertation without accurate data. Once the data is collected, statistics can be produced based on that data. Statistical consultants can then interpret the data so that the student can apply it properly to his or her dissertation. Statistical consultants, then, can be the missing link in a student’s quest to obtain his or her dissertation.

Statistical consultants are essential to anyone needing to work with statistics. Additionally, statistical consultants are cost effective. Compared to the time and money that would be spent attempting to get accurate statistics, hiring a statistical consultant is not expensive.
It is important, however, that the person hiring the statistical consultants choose the right statistical consulting firm. Proper research must be done on the statistical consulting firm that is obtained. This research does not take long and essentially ensures that the statistical consultants will provide what the client needs. This research can be as simple as inquiring if the statistical consultant has a PhD. The reason it is important to hire a statistical firm with a PhD is because a PhD knows how to properly obtain and interpret statistics. Additionally, if you are a student and you need to write a dissertation, someone who has been there and done it before (a PhD) can prove invaluable as you face the same challenges and hardships that the PhD has faced in the past. In other words, because the PhD has “been there and done that,” they know the unique circumstances of writing a dissertation. Thus, their help can be all the more precise and informative. For information on Statistics Solutions and the PhD qualifications, click here.

Whatever your statistical needs, statistical consultants can provide easy, effective, and inexpensive solutions to your statistical questions and problems.

Logistic Regression

Logistic regression is an extension of multiple linear regressions, where the dependent variable is binary in nature. Logistic regression predicts the discreet outcome, such as group membership, from a set of variables that may be continuous, discrete, dichotomous, or of any other type. Logistic regression is an extension of discriminant analysis. Discriminant analyses also predict the group memberships of the dependent variable, much like logistic regression. However, in discriminant analysis, there is an assumption of the relationship between the normal and linear distribution. Often, assumptions of equal variance do not meet. But in logistic regression, there is no assumption related to normal distribution, linear relationship and equal variance. In logistic regression, there may be many independent variables, like multiple-linear regressions.

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The model:

In logistic regression, the dependent variable is dichotomous. In logistic regression, we can take the value of 1 with the probability of success q and or the value 0, with the probability of failure 1- q. When there are two dependent variable categories, then it is said to be binary logistic regression. When there are more than two dependent variable categories, then it is a form of multinomial logistic regression. Symbolically, the probability of the dependent variable can be measured by using the following formula:






Where α= the constant of the equation and β= the coefficient of the predictor variable. An alternative form of logistic regression can be represented as the following:





Logistic regression has two main uses. The first use of logistic regression is that it predicts group membership. Second, logistic regression tells us about the relationship and strengths among the variables.

Test statistics in logistics:

1. Wald statistics: In logistic regression, Wald statistics is used to test the significance of each variable. In logistic regression, Wald statistics is simply the Z statistics, which is simply described as the following:





After squaring the Z value, it follows the chi-square distribution. In the case of a small sample size, the likelihood ratio test is more suitable than Wald statistics in logistic regression.

2. Likelihood ratio: The Likelihood ratio test maximizes the value of the likelihood function for the full model. Symbolically it is as follows:




After the log transformation, the likelihood ratio test follows the chi-square distribution. In logistic regression, it is suggested that the likelihood ratio test is used for significance when we are using backward stepwise elimination.

3. Goodness of fit: In logistic regression, goodness of fit is measured by the Hosmer-lemshow test statistics. This statistic basically compares the observed and predicted observation for the goodness of fit model.

Logistic regression and statistical software: Most software, like SPSS, STATA, SAS, and MATLAB, etc. have the option of performing logistic regression. In SAS, there is a procedure to perform logistic regression. SPSS is GUI software and it has the option to perform logistic regression. To perform logistic regression in SPSS, select the analysis menu from SPSS and select “binary logistic regression” from the regression option. If the dependent variable has more than two categories, then select the “multinomial model” from the regression option. If data are in order, then select the “ordinal logistic regression” from the regression option. After clicking on the logistic regression, select “binary variable” as the dependent variable, “others” as the continuous variables and “dichotomous variable” as the independent variable. After selecting the dependent and independent variable, select the model for logistic regression. The user can select to see both backward and forward methods in logistic regression.

Dispersion

In statistics, the measure of central tendency gives a single value that represents the whole value. But the central tendency cannot describe the observation fully. The measure of dispersion helps us to study the variability of the items. In a statistical sense, dispersion has two meanings: first it measures the variation of the items among themselves, and second, dispersion measures the variation around the average. If the difference between the value and average is high, then dispersion will be high. Otherwise it will be low. According to Dr. Bowley, “dispersion is the measure of the variation between items.” Researchers use the technique of dispersion because it determines the reliability of the average. Dispersion also helps a researcher in comparing two or more series. Dispersion is also the facilitating technique to many other statistical techniques like correlation, regression, structural equation modeling, etc. In statistics, dispersion has two measure types. The first is the absolute measure, which measures the dispersion in the same statistical unit. The second type of dispersion is the relative measure of dispersion, which measures the dispersion in a ratio unit. In statistics, there are many techniques that are applied to measure dispersion.

Range: Range is the simple measure of dispersion, which is defined as the difference between the largest value and the smallest value. Mathematically, the absolute and the relative measure of range can be written as the following:

R= L - S

Where R= Range, L= largest value, S=smallest value

Quartile deviation: This is a measure of dispersion. In this method, the difference between the upper quartile and lower quartile is taken and is called the interquartile range. Symbolically it is as follows:

Where Q3= Upper quartile Q1= Lower quartile

Mean Deviation: Mean deviation is a measure of dispersion, which is known as the average deviation. Mean deviation can be computed from the mean or median. Mean deviation is the arithmetic deviation of different items of central tendency. It may be the mean or the median. Symbolically, mean deviation is defined as the following:

Where M= median,

= mean

Standard Deviation: In the measure of dispersion, the standard deviation method is the most widely used method. In 1983, it was first used by Karl Pearson. Standard deviation is also known as root mean square deviation. Symbolically it is as follows:

Where

=Deviation

= standard deviation N= total number of observations.

Variance: Variance is another measure of dispersion. The term variance was first used in 1918, by R.A Fisher. Variance is known as the square of the standard deviation. Symbolically, variance can be written as the following:

Variance= (S.D)

2 =

If we know standard deviation, then we can compute the variance by squaring it. If we have variance, then we can also compute the standard deviation, by using the following formula:

Standard deviation has some mathematical properties. They are as follows:

1. Standard deviation of the n natural numbers can be found by using the following formula:

2. The sum of the square deviation taken by the arithmetical mean is minimal.

3. In asymmetrical distribution, Standard deviation has the following relationship with the mean

, Including 68.27% of the items

Include 95.45% of the items

Include the 99.73% of the items.

Coefficient of variation: Coefficient of variation is the relative measure of the dispersion. This method was developed by Karl Pearson. Coefficient of variation is used, while measuring the dispersion of two series. Coefficient of variation can be calculated by using the following formula:

Where C.V. = coefficient of variance, = standard deviation and = mean.

Discreet Probability Distribution

Discreet probability distribution is that type of probability distribution that refers to the discreet type of random variables that deal with some different kinds of discreet probability distribution.

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A uniform distribution in discreet probability distribution mainly consists of the range that is basically from [1, n]. This discreet probability distribution has a probability mass function (pmf) which is given by the following:

P(X=x)=1/n, for x = 1 …. n. This type of discreet probability distribution has a parameter that is denoted by ‘n.’ This type of discreet probability distribution generally belongs to the set of all positive integers. This type of discreet probability distribution is basically applicable in cases where an experiment consisting of throwing dice or a deck of cards, etc. is involved. In this discreet probability distribution, X is the random variable.

A Bernoulli distribution is also a type of discreet probability distribution, and it consists of the parameter ‘p’ and has the probability mass function (pmf) of P(X=x)= px (1-p)1-x, for x = 0,1. In this discreet probability distribution, X is the random variable.

In this type of discreet probability distribution, the parameter ‘p’ mainly satisfies the two types of values, i.e. 0 and 1. This discreet probability distribution is applicable in cases where the outcomes are dichotomous in nature, i.e. either success or failure.

The discreet probability distribution, called binomial distribution, assumes the non negative values and its probability mass function (pmf) is given by the following:

P(X=x)=(ncx) pxqn-x, x = 0,1 …. n;q=1-p.

This discreet probability distribution mainly deals with cases like the tossing of a coin.
The discreet probability distribution, called Poisson distribution, has the probability mass function (pmf) that is as follows:

P(X=x)=e-α αx/x!, x=0,1,…… where X in the pmf of this discreet probability distribution is nothing but the random variable, and α is the parameter.

This type of discreet probability distribution is mainly applicable in cases where one wants to find the number of faulty blades in a packet of one hundred blades, or one wants to determine the number of suicides reported in a particular city, etc. The population of the analysis acquires this type of discreet probability distribution and can be used in cases where one wants to determine things such as the number of printing mistakes on each page of a book, the number of cars passing a crossing during the busy hours of a day, the number of airplane accidents in some unit of time, the emission of radioactive (alpha) particles, etc.

The discreet probability distribution, called geometric distribution, has the probability mass function (pmf) that is given by the following:

P(X=x)=qx p; x=0,1, …..

This discreet probability distribution is generally applicable in cases that consist of a series of independent trials, and where the probability of success, which is represented by ‘p,’ is generally constant.

The discreet probability distribution, called hyper geometric distribution, has the probability mass function (pmf) which is given as the following:

P(X=x)= Mck N-Mcn-k / Ncn ; k= 0 , 1 , ….. , min(n,M).

The pmf of this discreet probability distribution consists of the parameters namely ‘N,’ ‘M,’ and ‘n,’ and these parameters are positive integers. This discreet probability distribution is applicable in experiments such as an experiment where the drawing of balls by means of simple random sampling is done without the replacement.

The discreet probability distribution, called multinomial distribution, is basically nothing but the generalization of the binomial distribution whose probability mass function (pmf) is given by the following:

p(x1, x2, …. , xk)=(n!/ x1!, x2!, …. , xk!) p1x1, p2x2, ….. , pkxk , ∑xi = n and ∑pi=1, i=1, .. , n.

This type of discreet probability distribution is applicable in those kinds of experiments where there is the ‘n’ repeated trials in which each number of trials has the possibility of having a discreet number of outcomes.

Attribute

An attribute basically indicates the quality of characteristics. The theory of attributes basically deals with qualitative types of characteristics that are calculated by using quantitative measurements. Therefore, the attribute needs slightly different kinds of statistical treatments, which the variables do not get. Attributes refer to the characteristics of the item under study, like the habit of smoking, or drinking. So ‘smoking’ and ‘drinking’ both refer to the example of an attribute.

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The researcher should note that the techniques involve statistical knowledge and are used at a wider extent in the theory of attributes.

In the theory of attributes, the researcher puts more emphasis on quality (rather than on quantity). Since the statistical techniques deal with quantitative measurements, qualitative data is converted into quantitative data in the theory of attributes.

There are certain representations that are made in the theory of attributes. The population in the theory of attributes is divided into two classes, namely the negative class and the positive class. The positive class signifies that the attribute is present in that particular item under study, and this class in the theory of attributes is represented as A, B, C, etc. The negative class signifies that the attribute is not present in that particular item under study, and this class in the theory of attributes is represented as α, β, etc.

The assembling of the two attributes, i.e. by combining the letters under consideration (such as AB), denotes the assembling of the two attributes.

This assembling of the two attributes is termed dichotomous classification. The number of the observations that have been allocated in the attributes is known as the class frequencies. These class frequencies are symbolically denoted by bracketing the attribute terminologies. (B), for example, stands for the class frequency of the attribute B. The frequencies of the class also have some levels in the attribute. For example, the class that is represented by the ‘n’ attribute refers to the class that has the nth order. For example, (B) refers to the class of 2nd order in the theory of attributes.

These attribute symbols also play the role of an operator. For example, A.N=(A) means that the operation of dichotomizing N according to the attribute A gives the class frequency equal to (A).
There is also independence nature in the theory of attributes. The two attributes are said to be independent only if the two attributes are absolutely uncorrelated to each other.

In the theory of attributes, the attributes A and B are said to be associated with each other only if the two attributes are not independent, but are related to each other in some way or another.

The positive association in the two attributes exists under the following condition:

(AB) > (A) (B)/ N.

The negative association in the two attributes exists under the following condition:

(AB) < (A) (B) /N.

The situation of complete association in the two attributes arises when the occurrence of attribute A is completely dependent upon the occurrence of attribute B. However, attribute B may occur without attribute A, and the same thing holds true if attribute A is the independent one.

Ordinarily, the two attributes are said to be associated if the two occur together in a number of cases.

The consistency between the two attributes (A)=20 and (AB)=25 is not present as the attribute (AB) cannot be greater than attribute (A) if they have been observed from the same population.

Runs Test of Randomness

Run test of randomness is a statistical test that is used to know the randomness in data. Run test of randomness is sometimes called the Geary test, and it is a nonparametric test. Run test of randomness is an alternative test to test autocorrelation in the data. Autocorrelation means that the data has correlation with its lagged value. To confirm whether or not the data has correlation with the lagged value, run test of randomness is applied. In the stock market, run test of randomness is applied to know if the stock price of a particular company is behaving randomly, or if there is any pattern. Run test of randomness is basically based on the run. Run is basically a sequence of one symbol such as + or -. Run test of randomness assumes that the mean and variance are constant and the probability is independent.

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Procedure for run test for randomness:

Hypothesis: To test the run test of randomness, first set up the null and alternative hypothesis. In run test of randomness, null hypothesis assumes that the distributions of the two continuous populations are the same. The alternative hypothesis will be the opposite of the null hypothesis.

Calculation of statistics: In the run test of randomness, the second step is the calculation of the mean and variance. The mean and variance in run test of randomness is calculated by using the following formula:

Mean:

Variance:

Where N= Total number of observations =N1+N2
N1=Number of + symbols
N2=Number of – symbols
R= number of runs
If the run test of randomness is sustainable with the null hypothesis, then we can expect the following properties of normal distribution:

Decision run, in run test of randomness: If the calculated value of the run test of randomness lies within the preceding confidence interval, then do not reject the null hypothesis. If the calculated value of the run test of randomness lies outside the preceding confidence interval, then reject the null hypothesis.

Assumptions in run test of randomness:

1. Data level: In run test of randomness, it is assumed that the data is recorded in order and not in a group. If data is not in order, then we have to assign the mean, median or mode value to the data.

2. Data Scale: In run test of randomness it is assumed that data is in numeric form. This condition is compulsory in run test of randomness, because in numeric data, it is easy to assign run to the numeric data.

3. Distribution: Run test of randomness is a nonparametric test, so it does not assume any assumption about the distribution.

4. In run test of randomness, the probability of run is independent.

Run test of randomness and SPSS: These days, statistical software makes very easy calculations of the run test. Statistical software performs the run test of randomness. In SPSS, run test of randomness can be performed by selecting the “run test” option from the nonparametric options available in the analysis menu. As we select the run test option, a window appears with the variable list. Select the variable for the run test from this window and drag it into the test variable list. If data is not in order, then select the “cut if” point. Select the “significance level” and “descriptive statistics,” from the option menu. After selecting these options, click on the “ok” button. Results of the run test of randomness appear in the SPSS result window. In SPSS, the output probability value is used for making the decision of whether we are going to accept or reject the null hypothesis. If the probability value of the run test of randomness is greater than the predetermined significance value, then we will accept the null hypothesis. If the calculated probability value is less than the predetermined significance value, then we will reject the null hypothesis.

Autocorrelation

Autocorrelation in statistics is a mathematical tool that is usually used for analyzing functions or series of values, for example, time domain signals. In other words, autocorrelation determines the presence of correlation between the values of variables that are based on associated aspects. In a way, it is the cross-correlation of a signal with itself. Most of the statistical models are based upon the assumption of instance independence, which gets desecrated by autocorrelation. Autocorrelation is very useful in activities where there are repeated signals. Autocorrelation is used to determine the periodic signals that get obscured beneath the noise, or it is used to realize the fundamental frequency of a signal that does not have that frequency as a component but applies it there along with many harmonic frequencies.

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In statistics, the correlation of a method against a time-shifted version is the autocorrelation of a discrete time series method. There are many applications where autocorrelation is very useful. Autocorrelation is used for the measurement of the optical spectra, and for the measurements of minutely lived light pulses that are produced by lasers. This is done with the help of optical autocorrelators. Autocorrelation is used in optics, where the normalized autocorrelations and cross correlations together give the degree of coherence of an electromagnetic field. Autocorrelation is also used for signal processing. Autocorrelation can help you get information about repetitive events like musical beats or pulsar frequencies, but it can still not give the position in time of the beats. Thus, autocorrelation can be used for identifying non-randomness in data and to classify an appropriate time series model for the non random data.

In other words it can be said that autocorrelation is a correlation coefficient, where the correlation is not between the two different variables. Rather, it is between the different values of the same variables. When autocorrelation is used to spot non–randomness, it is mostly the autocorrelation of the first lag that is taken into consideration. Autocorrelation used to determine a proper time series model is conventionally plotted for various lags.


Thus, we can say that autocorrelation is useful for answering two main questions:

1. Was what was generated taken from a random process or a non–random process?
2. Which model is a more appropriate model for the generated data?


Autocorrelation has many properties and it is used for diverse studies and diverse dimensions. The properties of autocorrelation are interchangeable in different dimensions. The most important quality of autocorrelation is symmetry and consistency. Autocorrelation for periodic functions is also periodic, with a similar period. If the functions are completely uncorrelated, then their autocorrelation is the sum of the autocorrelations of each function separately. Autocorrelation is a unique kind of cross correlation, and thus it bears all the properties of cross correlation. The autocorrelation function is easily available is general statistical software programs, thus one can easily access autocorrelation.

ANOVA

ANOVA stands for analysis of variance and was developed by Ronald Fisher in 1918. Thus, some researchers also call it the statistics fisher analysis of variance. ANOVA is a statistical method that is used to do the analysis of variance between n groups. T-test is the test used when the researcher wants to compare two groups. For example, if a researcher wants to compare the income of people based on their gender, they can use the t-test. Here, we have two groups: male and female. To compare the two groups, T-test is the best test. However, there is sometimes a problem when comparing groups that are more than two groups. In these cases, when we want to compare more than two groups, we can use the T-test as well, but this procedure is long. For example, first we would have to compare the first two groups. Then we would have to compare the last two groups. Finally, we would have to compare the first and the last group. This would take more time and there would be more possibility for mistakes. Thus, Fisher developed a test called ANOVA that can be applied to compare the variance when groups are more than two. ANOVA statistics also belong to the parametric test family, because ANOVA has some assumptions. When data meets these specific assumptions, then ANOVA is a more powerful test than the nonparametric test.

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Assumptions in ANOVA:

1. Normality: The first assumption in ANOVA is that the data should be normally distributed or the distribution of the particular data should be normal. There are many statistical tests that are applied to know the distribution of the ANOVA data. More commonly, many researchers use Kolmogorov-Smirnov, Shapiro-Wilk, or the histogram test to test the normality of the data.

2. Homogeneity: The second important assumption in ANOVA is that homogeneity or variance between the groups should be the same. In SPSS, Levene’s test is applied to test the homogeneity of the ANOVA data.

3. The third assumption is ANOVA is independence of case. This means that the grouping variables should be independent of each other or there should not be any pattern between the cases.

In research, after the regression technique, ANOVA is the second technique that is the most commonly used by the researcher. It is used in business, medicine or in psychology research. For example, in business, ANOVA is used to know the sales difference of different regions. A Psychology researcher can use ANOVA to compare the behavior of different people. A medical researcher can use ANOVA statistics in the experiment of a drug as he or she can test whether or not the drug cures the illness.

Procedure of ANOVA:

Set up hypothesis: To perform ANOVA statistics, a researcher has to set up the null and alternative hypothesis.

Calculation of MSB, MSW and F ratio: After set up, the researcher must calculate the hypothesis and the variance between the samples. In the calculation of variance between the samples, first we calculate the grand mean from the all the samples. Then, the researcher must make the deviation from individual mean to the grand mean for each sample, and square the deviation and divide the square deviation of all the samples by their degree of freedom. This is called MSB, which stands for the mean sum of square between the samples. The second component of ANOVA statistics will be the variance within the sample. To calculate the variance within the sample, take each deviation sample from the respective sample means, find the square of each sample, and divide it by the respective degree of freedom. This is called MSW, which stands for the mean sum of the square within the sample. The ratio of the MSB and MSW is called the F ratio.

Testing of hypothesis in ANOVA: In ANOVA statistics, the calculated F ratio value is compared to the standardized table value. If the calculated F ratio value is greater than the table value, we will reject the null hypothesis and conclude that the means of the groups are different. If the calculated value is less than the table value, then we will accept the null hypothesis and conclude that the means of all the groups are the same.

ANOVA and SPSS: Manual calculation of ANOVA statistics is a long procedure. These days, almost all statistical computer software has the option for calculating ANOVA statistics. In SPSS, ANOVA can be performed by using the “analysis menu” and the “compare means option.” Select “one way ANOVA” from the compare means option. In SPSS, the hypothesis probability value is used to accept or reject the null hypothesis.

Statistics Consulting

The benefits of statistics consulting cannot be overstated, for statistics consulting can be beneficial for many individuals and groups, across many fields and disciplines. In fact, statistics consulting has become more and more popular as the demand to keep up with the fast-paced and ever-changing world in which we live increases. Because there is more research available and more access to information than ever before, statistics consulting can play a major role in shaping a business, providing results for the medical field, or assisting a student who must complete his or her dissertation.

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In business, statistics consulting can be particularly useful as statistics consulting can provide information that can maximize profits. This is true because statistics consulting can ‘crunch’ the numbers and data of what works for a business and what does not work. Statistics consulting, then, can efficiently direct a business towards what works and this can save valuable time and money for the business or cooperation. Additionally, statistics consulting can provide market research for a business and this can give the business valuable information as to what will work in the future. Thus, a business can prevent the very pricey launching of a service or product that will not work. Instead, statistics consulting will provide data on market research and statistics consulting can ensure that the business launches the right products in the right market.

Statistics consulting can be especially useful for small businesses. This is true because oftentimes, small businesses do not have the man-power or statistical know-how to carry out statistics and market research properly. Statistics consulting can be obtained by these small businesses and this can save that small business a lot of money in the long-term. Thus, with the help of statistics consulting, small businesses can compete with larger ones.

Statistics consulting is also very useful to the medical field. This is true because the medical field needs to know what drugs work, at what dosage, etc. And while the doctors are the ones who prescribe the medication and do research on the actual medication, a statistician is called in to look at the results of the research and to set the parameters of the research. This is because doctors are trained in the human body, not in the complex world of statistics. Statistics consulting provides the ‘missing link’ to these doctors and armed with the information and the statistics that statistics consulting provides, doctors can make appropriate decisions as to what medicines are effective at what doses.

And while statistics consulting is very beneficial to both the business world and the medical world, statistics consulting is perhaps most useful to students who need to write their dissertation. This is because the dissertation requires statistics to be successful. And while the student who is seeking his or her doctoral degree is certainly well qualified to speak on his or her subject matter, oftentimes they are not prepared or trained to do the complex statistical procedures needed for their dissertations. Thus, statistics consulting can assist these students as they complete the necessary statistics and research to obtain their dissertations.
Clearly then, the benefits of statistics consulting cannot be overstated as they provide the statistical know-how to complete any statistics problem. Whether it is a business looking to maximize profit, a doctor or hospital needing to rely on statistical medical research, or a student seeking help on his or her dissertation, statistics consulting is the solution to any and all of these fields. There is absolutely no question that statistics consulting will ensure success for anyone who needs help with statistics.

Statistical Help

If you are struggling with statistics, it is time to get statistical help. And while it might not be easy for you to admit that you actually need help, once you seek statistical help, you will be very glad you have done so, for statistical help can assist you with any statistical issue you might be having.

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Statistical help is available to anyone who needs help with statistics (hence the name statistical help). Because statistics is a science, it requires that precise methodology be followed and adhered to. For people who are not trained in statistics (and who don’t seek statistical help), this can be very difficult. These people who are not trained statisticians and need statistical help sometimes turn to the internet for their statistical help. And while this is a good instinct, for seeking help and trying to do it on your own is very good in some cases, in the case of statistics, this can often have disastrous consequences.

Let’s say, for example, a student who needs statistical help turns to the internet for statistical help. That student reads-up on statistics and spends hours and hours researching the proper methodology for his or her project. Well, in reality, the study of statistics takes weeks and months, if not years to perfect. In fact, entire courses are designed in statistics. Thus, quickly reading and researching information about statistical procedures on the internet will not provide a student with the statistical help he or she needs.

Continuing with this example, let’s say this student, armed with hours of internet research on statistics, continues with his or her dissertation by conducting the necessary research. This student (who has not acquired proper statistical help) then gathers data. And again, while the practice of using the intent can sometimes be good (the student wanting to do it on his or her own, again, is a good instinct) the outcome can be very painful. For it is likely that a student who is not trained in statistics and who does not seek appropriate statistical help will falter in the data gathering step of statistics. And this step is extremely important if one is to obtain accurate and valid statistics. That student then relies on faulty data (data that is biased, for example, or data that has been obtained with an improper sample size), and this faulty data leads to faulty conclusions and inferences. Thus, because this student did not get the right kind of statistical help, he or she must start all over. And there is nothing more time-consuming and frustrating than starting a dissertation from the very beginning.

Instead of trying to do all of the statistics on his or her own, that student should turn to statistics consultants for statistical help. Statistics consultants can give statistical help thereby ensuring that the student is performing the proper statistical methodology. Additionally, statistics consultants can ensure that the student has accurate and precise results. Statistical help, then, can step-in to help and assist the student so that he or she does not make mistakes in the statistical portion of their analysis.

Statistical help provided by statistical consultants provides effective, efficient and accurate results. This is because statistical help is provided by trained and expert statistical consultants who know everything there is to know about statistics. The experts who provide statistical help know, for example, how to gather accurate data, how to interpret that data, how to use statistical tests to accurately read that data, and how to apply those results to a dissertation or project.

Statistical help is also very affordable. This is especially true when considering the time that is often spent making mistakes and starting over because of a statistical procedural mistake. Statistical help provides accurate and on-time results and for this reason, statistical help is well worth the investment. And while the process of statistics can be time-consuming and laborious, with the right statistical help, a student can finish with success.

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