Latent class analysis (LCA) is a multivariate technique that can be applied for cluster, factor, or regression purposes.
Latent class analysis (LCA) is commonly used by the researcher in cases where it is required to perform classification of cases into a set of latent classes. The Latent class analysis (LCA) carried out on latent classes are based on categorical types of indicator variables. In Latent class analysis (LCA), indicator variables are those variables that are assigned as ‘1’ if their condition is true, and are otherwise assigned as ‘0.’
Latent class analysis (LCA) uses a variant called Latent profile analysis for continuous variables. Mixture modeling with the structural equation models is a major type of Latent class analysis (LCA).
Latent class analysis (LCA) divides the cases into latent classes that are conditionally independent. In other words, Latent class analysis (LCA) divides those cases in which the variables of interest are not correlated within any other variables in the class.
The model parameters in Latent class analysis (LCA) are the maximum likelihood estimates (MLE) of conditional response probabilities.
There are two ways by which the number of the latent classes in the Latent class analysis (LCA) is determined. The first and more popular method is to perform an iterative test of goodness of fit models with the latent classes in Latent class analysis (LCA) using the likelihood ratio chi square test.
The other method is the method of bootstrapping of the latent classes in Latent class analysis (LCA). The rho estimates refer to the item response probabilities in Latent class analysis (LCA).
The odds ratio in Latent class analysis (LCA) measures the effective sizes of the covariates in the model. The odds ratio in Latent class analysis (LCA) is calculated by carrying out multinomial regression. The dependent variable in this regression in Latent class analysis (LCA) is the latent class variable, and the independent variable is the covariate.
If the value of the odds ratio in Latent class analysis (LCA) is 1.5 for class 1, then it means that a unit increase in the covariate corresponds to a 50 % greater likelihood.
The posterior probabilities in Latent class analysis (LCA) refer to the probability of that observation that is classified in a given class.
Latent class analysis (LCA) is done using software called Latent Gold. This software in Latent class analysis (LCA) implements Latent class models for cluster analysis, factor analysis, etc. The latent models in Latent class analysis (LCA) support nominal, ordinal as well as continuous data.
There are certain measures of model fit in Latent class analysis (LCA).
The latent model in Latent class analysis (LCA) can be fitted to the data with the help of likelihood ratio chi square. The larger the value of the statistic in Latent class analysis (LCA), the more inefficient the model is to fit the data.
The difference chi square in Latent class analysis (LCA) is used to calculate the difference of the two model chi squares for the two nested models.
In order to assess the validity or the reliability of Latent class analysis (LCA) a statistic called Cressie-Read statistic is used. The validity of Latent class analysis (LCA) can be assessed with the help of the probability value being compared with the probability value of the model chi square.
It is assumed that Latent class analysis (LCA) does not follow linearity within the data.
Latent class analysis (LCA) does not follow the normal distribution of the data.
Latent class analysis (LCA) does not follow the homogeneity of variances.
Tuesday, May 26, 2009
Hypothesis Testing
Hypothesis testing is a scientific process of testing whether or not the hypothesis is plausible.
The following steps are involved in hypothesis testing:
The first step in hypothesis testing is to state the null and alternative hypothesis clearly. The null and alternative hypothesis in hypothesis testing can be a one tailed or two tailed test.
The second step in hypothesis testing is to determine the test size. This means that the researcher decides whether a test should be one tailed or two tailed to get the right critical value and the rejection region.
The third step in hypothesis testing is to compute the test statistic and the probability value. This step of the hypothesis testing also involves the construction of the confidence interval depending upon the testing approach.
The fourth step in hypothesis testing involves the decision making step. This step of hypothesis testing helps the researcher reject or accept the null hypothesis by making comparisons between the subjective criterion from the second step and the objective test statistic or the probability value from the third step.
The fifth step in hypothesis testing is to draw a conclusion about the data and interpret the results obtained from the data.
There are basically three approaches to hypothesis testing. The researcher should note that all three approaches require different subject criteria and objective statistics, but all three approaches of hypothesis testing give the same conclusion.
The first approach of hypothesis testing is to test the statistic approach.
The common steps in all three approaches of hypothesis testing is the first step, which is to state the null and alternative hypothesis.
The second step of the test statistic approach of hypothesis testing is to determine the test size and to obtain the critical value. The third step of the test statistic approach of hypothesis testing is to compute the test statistic. The fourth step of the test statistic approach of hypothesis testing is to reject or accept the null hypothesis depending upon the comparison between the tabulated value and the calculated value. If the tabulated value in hypothesis testing is more than the calculated value, than the null hypothesis is accepted. Otherwise it is rejected. The last step of this approach of hypothesis testing is to make a substantive interpretation.
The second approach of hypothesis testing is the probability value approach. The second step of this approach in hypothesis testing is to determine the test size. The third step of this approach of hypothesis testing is to compute the test statistic and the probability value. The fourth step of this approach of hypothesis testing is to reject the null hypothesis if the probability value is less than the tabulated value. The last step of this approach of hypothesis testing is to make a substantive interpretation.
The third approach of hypothesis testing is the confidence interval approach. The second step of hypothesis testing is to determine the test size or the (1-test size) and the hypothesized value. The third step of hypothesis testing is to construct the confidence interval. The fourth step of hypothesis testing is to reject the null hypothesis if the hypothesized value does not exist in the range of the confidence interval. The last step of this approach of hypothesis testing is to make the substantive interpretation.
The first approach of hypothesis testing is a classical test statistic approach, which computes a test statistic from the empirical data and then makes a comparison with the critical value. If the test statistic in this classical approach of the hypothesis testing is larger than the critical value, then the null hypothesis is rejected. Otherwise, it is accepted.
The following steps are involved in hypothesis testing:
The first step in hypothesis testing is to state the null and alternative hypothesis clearly. The null and alternative hypothesis in hypothesis testing can be a one tailed or two tailed test.
The second step in hypothesis testing is to determine the test size. This means that the researcher decides whether a test should be one tailed or two tailed to get the right critical value and the rejection region.
The third step in hypothesis testing is to compute the test statistic and the probability value. This step of the hypothesis testing also involves the construction of the confidence interval depending upon the testing approach.
The fourth step in hypothesis testing involves the decision making step. This step of hypothesis testing helps the researcher reject or accept the null hypothesis by making comparisons between the subjective criterion from the second step and the objective test statistic or the probability value from the third step.
The fifth step in hypothesis testing is to draw a conclusion about the data and interpret the results obtained from the data.
There are basically three approaches to hypothesis testing. The researcher should note that all three approaches require different subject criteria and objective statistics, but all three approaches of hypothesis testing give the same conclusion.
The first approach of hypothesis testing is to test the statistic approach.
The common steps in all three approaches of hypothesis testing is the first step, which is to state the null and alternative hypothesis.
The second step of the test statistic approach of hypothesis testing is to determine the test size and to obtain the critical value. The third step of the test statistic approach of hypothesis testing is to compute the test statistic. The fourth step of the test statistic approach of hypothesis testing is to reject or accept the null hypothesis depending upon the comparison between the tabulated value and the calculated value. If the tabulated value in hypothesis testing is more than the calculated value, than the null hypothesis is accepted. Otherwise it is rejected. The last step of this approach of hypothesis testing is to make a substantive interpretation.
The second approach of hypothesis testing is the probability value approach. The second step of this approach in hypothesis testing is to determine the test size. The third step of this approach of hypothesis testing is to compute the test statistic and the probability value. The fourth step of this approach of hypothesis testing is to reject the null hypothesis if the probability value is less than the tabulated value. The last step of this approach of hypothesis testing is to make a substantive interpretation.
The third approach of hypothesis testing is the confidence interval approach. The second step of hypothesis testing is to determine the test size or the (1-test size) and the hypothesized value. The third step of hypothesis testing is to construct the confidence interval. The fourth step of hypothesis testing is to reject the null hypothesis if the hypothesized value does not exist in the range of the confidence interval. The last step of this approach of hypothesis testing is to make the substantive interpretation.
The first approach of hypothesis testing is a classical test statistic approach, which computes a test statistic from the empirical data and then makes a comparison with the critical value. If the test statistic in this classical approach of the hypothesis testing is larger than the critical value, then the null hypothesis is rejected. Otherwise, it is accepted.
Content Analysis
Content analysis provides information related to newspaper promotions, stories, photographs, displays and classified advertisements, etc. All of these fall in the scope of the study of Content analysis. Content analysis consists of counting and classifying content.
For a free consultation on content analysis, click here.
In Content analysis, the researcher gathers information about the entire paper. This information includes the number of pages, the number of sections, etc. Then, an in-depth Content analysis is performed by the researcher on each content item.
The researcher in Content analysis analyzes each story in terms of its attributes, including topics, sources treatments, writing styles, etc. The researcher in Content analysis analyzes newspaper promotions for things like type, color, topic and size.
The researchers conduct content analysis for certain reasons. These reasons are as follows:
There are certain terms that are used in content analysis that are helpful in understanding Content analysis. For example, unitizing in content analysis is a process in which the investigator establishes uniformity in the analysis. Thus, the researcher in content analysis unitizes the words, sentences, paragraphs, etc.
Sampling is one of the crucial weapons in content analysis. The sampling plan in Content analysis is designed to minimize the distortion caused in some particular content due to certain major events, etc. In Content analysis, the content is generally enormous. Thus, the researcher utilizes the technique of sampling in order to make his content in content analysis less complicated. The theory behind sampling in content analysis consists of counting. This involves development of different kinds of similar-meaning terms.
Inference is a major part of content analysis. A contextual phenomenon in content analysis must be analyzed in order to obtain a valid inference of the context for findings.
Content analysis involves conclusions that are usually communicated by the researcher in a narrative manner.
There are basically two assumptions in content analysis. First, content analysis is generally assumed to be subjected to the problems of sampling. Second, content analysis is assumed to be based upon the context for words and meanings.
There are certain software resources for conducting content analysis. These include the following:
For a free consultation on content analysis, click here.
In Content analysis, the researcher gathers information about the entire paper. This information includes the number of pages, the number of sections, etc. Then, an in-depth Content analysis is performed by the researcher on each content item.
The researcher in Content analysis analyzes each story in terms of its attributes, including topics, sources treatments, writing styles, etc. The researcher in Content analysis analyzes newspaper promotions for things like type, color, topic and size.
The researchers conduct content analysis for certain reasons. These reasons are as follows:
- Content analysis is usually carried out to study the discrepancy in the trends of the content with respect to time.
- Content analysis is carried out to describe the reasons why the readers focus on certain topics of the content.
- Content analysis can be used to make comparisons on international differences.
- Content analysis helps in comparing group differences in the content.
- Content analysis can expose the usage of biased terms in the research. Such biased terms can influence the opinions or behaviors of people.
- Content analysis is also useful in the testing of hypotheses about the cultural and symbolic usages of terms in the content.
- Content analysis helps the researcher for purposes of coding as well. Coding on open ended questions is done with the help of Content analysis.
There are certain terms that are used in content analysis that are helpful in understanding Content analysis. For example, unitizing in content analysis is a process in which the investigator establishes uniformity in the analysis. Thus, the researcher in content analysis unitizes the words, sentences, paragraphs, etc.
Sampling is one of the crucial weapons in content analysis. The sampling plan in Content analysis is designed to minimize the distortion caused in some particular content due to certain major events, etc. In Content analysis, the content is generally enormous. Thus, the researcher utilizes the technique of sampling in order to make his content in content analysis less complicated. The theory behind sampling in content analysis consists of counting. This involves development of different kinds of similar-meaning terms.
Inference is a major part of content analysis. A contextual phenomenon in content analysis must be analyzed in order to obtain a valid inference of the context for findings.
Content analysis involves conclusions that are usually communicated by the researcher in a narrative manner.
There are basically two assumptions in content analysis. First, content analysis is generally assumed to be subjected to the problems of sampling. Second, content analysis is assumed to be based upon the context for words and meanings.
There are certain software resources for conducting content analysis. These include the following:
- ATLAS.ti is used in content analysis as software for text analysis and model building.
- The General Inquirer is the classic package for content analysis.
- Intext and TextQuest is software developed by Harald Klein for content analysis.
Heteroscedasticity
The crucial assumption of a classical linear regression model is that the volatility that has occurred in the model should be uniform in nature. If this assumption is not satisfied by the model, then one would have to consider that the model has been exposed to heteroscedasticity.
There are examples that can be discussed to gain a better understanding of heteroscedasticity. In the case of an income expenditure model, if the income is decreased, then the expenditure will also simultaneously decrease, and vice versa. If, however, heteroscedasticity is present in the model, then as the income is increased, then the graph for the expenditure variable would remain constant.
For a free consultation on heteroscedasticity, click here.
Heteroscedasticity generally occurs due to the presence of an outlier. An outlier in relation to heteroscedasticity is nothing but an observation that is numerically apart from the rest of the observations given in the data.
Heteroscedasticity can occur if a major variable is eliminated from the model. In the case of the income expenditure model, for example, if the variable called ‘income’ is eliminated, then there would be no inference from that model, and one would have to consider that the model has undergone heteroscedasticity.
Heteroscedasticity can also occur due to the presence of symmetrical or assymeterical curves of the regressor included in the model.
Heteroscedasticity can also occur due to false data transformation and incorrect functional form (like comparisons between a linear model and a log linear model, etc.).
Heteroscedasticity is a common or popular type of disturbance, especially in cases involving cross sectional data or time series data. If investigators who conduct ordinary least squares (OLS) do not consider the disturbance caused by heteroscedasticity, then they would not be able to examine the confidence intervals and the tests of hypotheses. This is because in the presence of heteroscedasticity, the variance calculated would be significantly less than the variance of the best linear unbiased estimator. As a result, the outcomes of the significant tests will not be accurate due to heteroscedasticity.
For a researcher to detect the presence of heteroscedasticity in the data, certain informal tests have been proposed by several econometricians.
There is a high probability of heteroscedasticity in a cross sectional data if small, medium and large organizations are sampled together.
An informal method, called the graphical method, helps the researcher to detect the presence of heteroscedasticity. If the investigator assumes that there is no heteroscedasticity and then performs regression analysis, the estimated residuals (with the help of the graphical method) would then exhibit certain patterns that would indicate the presence of heteroscedasticity.
A formal test, called Spearman’s rank correlation test, is used by the researcher to detect the presence of heteroscedasticity.
Suppose the researcher assumes a simple linear model, for example- Yi = β0 + β1Xi + ui - to detect the presence of heteroscedasticity. The researcher then fits the model to the data by calculating the absolute values of the residual and further sorting them in ascending or descending manner to detect heteroscedasticity. Then, the researcher computes the value of Spearman’s rank correlation for heteroscedasticity.
The researcher then assumes the population rank correlation coefficient as zero and the size of the sample is assumed to be greater than 8 for heteroscedasticity. A significance test is carried out to detect heteroscedasticity. If the computed value of t is more than the tabulated value, then the researcher assumes that heteroscedasticity is present in the data. Otherwise heteroscedasticity is not present in the data.
There are examples that can be discussed to gain a better understanding of heteroscedasticity. In the case of an income expenditure model, if the income is decreased, then the expenditure will also simultaneously decrease, and vice versa. If, however, heteroscedasticity is present in the model, then as the income is increased, then the graph for the expenditure variable would remain constant.
For a free consultation on heteroscedasticity, click here.
Heteroscedasticity generally occurs due to the presence of an outlier. An outlier in relation to heteroscedasticity is nothing but an observation that is numerically apart from the rest of the observations given in the data.
Heteroscedasticity can occur if a major variable is eliminated from the model. In the case of the income expenditure model, for example, if the variable called ‘income’ is eliminated, then there would be no inference from that model, and one would have to consider that the model has undergone heteroscedasticity.
Heteroscedasticity can also occur due to the presence of symmetrical or assymeterical curves of the regressor included in the model.
Heteroscedasticity can also occur due to false data transformation and incorrect functional form (like comparisons between a linear model and a log linear model, etc.).
Heteroscedasticity is a common or popular type of disturbance, especially in cases involving cross sectional data or time series data. If investigators who conduct ordinary least squares (OLS) do not consider the disturbance caused by heteroscedasticity, then they would not be able to examine the confidence intervals and the tests of hypotheses. This is because in the presence of heteroscedasticity, the variance calculated would be significantly less than the variance of the best linear unbiased estimator. As a result, the outcomes of the significant tests will not be accurate due to heteroscedasticity.
For a researcher to detect the presence of heteroscedasticity in the data, certain informal tests have been proposed by several econometricians.
There is a high probability of heteroscedasticity in a cross sectional data if small, medium and large organizations are sampled together.
An informal method, called the graphical method, helps the researcher to detect the presence of heteroscedasticity. If the investigator assumes that there is no heteroscedasticity and then performs regression analysis, the estimated residuals (with the help of the graphical method) would then exhibit certain patterns that would indicate the presence of heteroscedasticity.
A formal test, called Spearman’s rank correlation test, is used by the researcher to detect the presence of heteroscedasticity.
Suppose the researcher assumes a simple linear model, for example- Yi = β0 + β1Xi + ui - to detect the presence of heteroscedasticity. The researcher then fits the model to the data by calculating the absolute values of the residual and further sorting them in ascending or descending manner to detect heteroscedasticity. Then, the researcher computes the value of Spearman’s rank correlation for heteroscedasticity.
The researcher then assumes the population rank correlation coefficient as zero and the size of the sample is assumed to be greater than 8 for heteroscedasticity. A significance test is carried out to detect heteroscedasticity. If the computed value of t is more than the tabulated value, then the researcher assumes that heteroscedasticity is present in the data. Otherwise heteroscedasticity is not present in the data.
Dissertation Statistics Consultation
A dissertation statistics consultation is a service that a dissertation statistics firm can provide to anyone who needs to write a dissertation. A dissertation statistics consultation is a way to help students who are struggling with the statistics part of their dissertation. Thus, dissertation statistics consultations can make the difficult task of writing a dissertation, and moreover, dealing with statistics, much more manageable.
To schedule a free dissertation statistics consultation, click here.
Dissertation statistics consultations involve closely working with a dissertation consultant. With the help of dissertation statistics consultations, the student can navigate the difficult aspects of his or her dissertation. And while a student’s advisor can provide similar help as a dissertation statistics consultation, oftentimes a student’s advisor is not easily accessible or available. Dissertation statistics consultations provided by dissertation consultants are always available however, as the main goal of the dissertation statistics consultation is to help students whenever they need help.
First, dissertation statistics consultations involve a lengthy discussion about the topic of study. In the dissertation statistics consultation, the student and the dissertation consultant discuss all aspects of the topic. The dissertation statistics consultation can provide valuable feedback to the student at this stage in the process, as the dissertation statistics consultation provided by the dissertation consultant will address several issues that are likely to come-up when the student attempts to get the topic approved. One such issue is whether or not the topic is appropriate and able to be studied. In other words, with the help of a dissertation statistics consultation, the student will have a better grasp of whether or not the topic can actually be studied, and whether or not that topic should be chosen by the student. Another issue that comes up during the topic- choosing phase is addressing the concern of whether or not that topic has been studied before. A proper dissertation statistics consultation will advise students on how to do research into whether or not their topic of study has already been studied.
The next service that a dissertation statistics consultation will provide is to explain to the student how to write that topic in a way that makes sense statistically. This part of the dissertation statistics consultation is essential because without the proper wording for the topic, the dissertation will not get approved. A dissertation statistics consultation will explain this wording so that it gets approved and makes sense to the student.
Once the topic is chosen and is phrased correctly, the dissertation statistics consultation provided by dissertation consultants can address the actual gathering of statistics. Because statistics is a science, the gathering of data and the interpretation of that data need to be done meticulously. A dissertation statistics consultation will ensure that the student knows how to gather proper statistics because a dissertation statistics consultation will discuss the methods, means and theories behind the gathering of data. Dissertation consultants are well versed in the gathering of data, and in the dissertation statistics consultation, the dissertation consultants will explain these precise methods of gathering data. Thus, with the help of dissertation consultants, students can gather data much quicker and much more efficiently.
Once the data is gathered, a dissertation statistics consultation provides all of the necessary information as to how to interpret the results and apply it to the dissertation. Here too the expertise of the dissertation consultant comes into play as the dissertation statistics consultation will go over every facet of the interpretation of results.
Finally, a dissertation statistics consultation service will ensure that the dissertation is finished accurately and on-time. With the on-going help of a dissertation consultant, the student will be able to finish his or her dissertation with much success.
To schedule a free dissertation statistics consultation, click here.
Dissertation statistics consultations involve closely working with a dissertation consultant. With the help of dissertation statistics consultations, the student can navigate the difficult aspects of his or her dissertation. And while a student’s advisor can provide similar help as a dissertation statistics consultation, oftentimes a student’s advisor is not easily accessible or available. Dissertation statistics consultations provided by dissertation consultants are always available however, as the main goal of the dissertation statistics consultation is to help students whenever they need help.
First, dissertation statistics consultations involve a lengthy discussion about the topic of study. In the dissertation statistics consultation, the student and the dissertation consultant discuss all aspects of the topic. The dissertation statistics consultation can provide valuable feedback to the student at this stage in the process, as the dissertation statistics consultation provided by the dissertation consultant will address several issues that are likely to come-up when the student attempts to get the topic approved. One such issue is whether or not the topic is appropriate and able to be studied. In other words, with the help of a dissertation statistics consultation, the student will have a better grasp of whether or not the topic can actually be studied, and whether or not that topic should be chosen by the student. Another issue that comes up during the topic- choosing phase is addressing the concern of whether or not that topic has been studied before. A proper dissertation statistics consultation will advise students on how to do research into whether or not their topic of study has already been studied.
The next service that a dissertation statistics consultation will provide is to explain to the student how to write that topic in a way that makes sense statistically. This part of the dissertation statistics consultation is essential because without the proper wording for the topic, the dissertation will not get approved. A dissertation statistics consultation will explain this wording so that it gets approved and makes sense to the student.
Once the topic is chosen and is phrased correctly, the dissertation statistics consultation provided by dissertation consultants can address the actual gathering of statistics. Because statistics is a science, the gathering of data and the interpretation of that data need to be done meticulously. A dissertation statistics consultation will ensure that the student knows how to gather proper statistics because a dissertation statistics consultation will discuss the methods, means and theories behind the gathering of data. Dissertation consultants are well versed in the gathering of data, and in the dissertation statistics consultation, the dissertation consultants will explain these precise methods of gathering data. Thus, with the help of dissertation consultants, students can gather data much quicker and much more efficiently.
Once the data is gathered, a dissertation statistics consultation provides all of the necessary information as to how to interpret the results and apply it to the dissertation. Here too the expertise of the dissertation consultant comes into play as the dissertation statistics consultation will go over every facet of the interpretation of results.
Finally, a dissertation statistics consultation service will ensure that the dissertation is finished accurately and on-time. With the on-going help of a dissertation consultant, the student will be able to finish his or her dissertation with much success.
Correlation
Correlation, as the name suggests, depicts a relationship between two or more variables under study. Correlation is generally categorized into two types, namely Bivariate Correlation and Partial Correlation.
For a free consultation on correlation or dissertation statistics, click here.
Bivariate Correlation is the one that shows an association between two variables. Correlation is the one that shows the association between two variables while keeping control or adjusting the effect of one or more additional variables.
A Correlation is a degree of measure, which means that a Correlation can be negative, positive, or perfect. A positive Correlation is a type of Correlation in which an increase changes the other variable. In other words, if there is an increase (or decrease) in one variable, then there is a simultaneous increase (decrease) in the other variable. A negative Correlation is a type of Correlation where if there is a decrease (or increase) in one variable, then there is a simultaneous increase (or decrease) in the other variables.
A perfect Correlation is that type of Correlation where a change in one variable affects an equivalent change in the other variable.
A British biometrician named Karl Pearson developed a formula to measure the degree of the Correlation, called the Correlation Coefficient. This Correlation Coefficient is generally depicted as ‘r.’ In mathematical language, the Correlation Coefficient, which was developed by the biometrician Karl Pearson, is defined as the ratio between the covariance of the two variables and the product of the square root of their individual variances. The range of the Correlation Coefficient generally lies between -1 to +1. If the value of the Correlation Coefficient is ‘+1,’ then the variable is said to be positively correlated. If, on the other hand, the value of the Correlation Coefficient is ‘-1,’ then the variable is said to be negatively correlated.
The value of the Correlation Coefficient does not depend upon the change in origin and the change in the scale.
If the value of the Correlation Coefficient is zero, then the variables are said to be uncorrelated. Thus, the variables would be regarded as independent. If there is no Correlation in the variables, then the change in one variable will not affect the change in the other variable at all, and therefore the variables will be independent.
However, the researcher should note that the two independent variables are not in any Correlation if the covariance of the variables is zero. This, however, is not true in the opposite case. This means that if the covariance of the two variables is zero, then it does not necessarily mean that the two variables are independent.
There are certain assumptions that come along with the Correlation Coefficient. The following are the assumptions for the Correlation Coefficient:
The Correlation Coefficient assumes that the variables under study should be linearly correlated.
Correlation coefficient assumes that a cause and effect relationship exists between different forces operating on the items of the two variable series. Such forces assumed by the correlation coefficient must be common to both series.
For the cases where operating forces are entirely independent, then the value of the correlation coefficient must be zero. If the value of the correlation coefficient is not zero, then in such cases, correlation is often termed as chance correlation or spurious correlation. For example, the correlation between the income of a person and the height of a person is a case of spurious correlation. Another example of spurious correlation is the correlation between the size of the shoe and the intelligence of a certain group of people.
A Pearsonian coefficient of correlation between the ranks of two variables, say, x and y, is called rank correlation coefficient between that group of variables.
For a free consultation on correlation or dissertation statistics, click here.
Bivariate Correlation is the one that shows an association between two variables. Correlation is the one that shows the association between two variables while keeping control or adjusting the effect of one or more additional variables.
A Correlation is a degree of measure, which means that a Correlation can be negative, positive, or perfect. A positive Correlation is a type of Correlation in which an increase changes the other variable. In other words, if there is an increase (or decrease) in one variable, then there is a simultaneous increase (decrease) in the other variable. A negative Correlation is a type of Correlation where if there is a decrease (or increase) in one variable, then there is a simultaneous increase (or decrease) in the other variables.
A perfect Correlation is that type of Correlation where a change in one variable affects an equivalent change in the other variable.
A British biometrician named Karl Pearson developed a formula to measure the degree of the Correlation, called the Correlation Coefficient. This Correlation Coefficient is generally depicted as ‘r.’ In mathematical language, the Correlation Coefficient, which was developed by the biometrician Karl Pearson, is defined as the ratio between the covariance of the two variables and the product of the square root of their individual variances. The range of the Correlation Coefficient generally lies between -1 to +1. If the value of the Correlation Coefficient is ‘+1,’ then the variable is said to be positively correlated. If, on the other hand, the value of the Correlation Coefficient is ‘-1,’ then the variable is said to be negatively correlated.
The value of the Correlation Coefficient does not depend upon the change in origin and the change in the scale.
If the value of the Correlation Coefficient is zero, then the variables are said to be uncorrelated. Thus, the variables would be regarded as independent. If there is no Correlation in the variables, then the change in one variable will not affect the change in the other variable at all, and therefore the variables will be independent.
However, the researcher should note that the two independent variables are not in any Correlation if the covariance of the variables is zero. This, however, is not true in the opposite case. This means that if the covariance of the two variables is zero, then it does not necessarily mean that the two variables are independent.
There are certain assumptions that come along with the Correlation Coefficient. The following are the assumptions for the Correlation Coefficient:
The Correlation Coefficient assumes that the variables under study should be linearly correlated.
Correlation coefficient assumes that a cause and effect relationship exists between different forces operating on the items of the two variable series. Such forces assumed by the correlation coefficient must be common to both series.
For the cases where operating forces are entirely independent, then the value of the correlation coefficient must be zero. If the value of the correlation coefficient is not zero, then in such cases, correlation is often termed as chance correlation or spurious correlation. For example, the correlation between the income of a person and the height of a person is a case of spurious correlation. Another example of spurious correlation is the correlation between the size of the shoe and the intelligence of a certain group of people.
A Pearsonian coefficient of correlation between the ranks of two variables, say, x and y, is called rank correlation coefficient between that group of variables.
Conjoint Analysis
Conjoint Analysis is considered a type of analyses that is used in the field of market research. Conjoint Analysis is used by the researcher to obtain the statistically significant importance that a consumer in the market gives to the crucial characteristics of a particular product. Conjoint Analysis is also used to obtain the utilities that a consumer in the market attaches to the various levels of the attributes of a particular product.
For a free consultation on conjoint analysis, click here.
All of this is determined in Conjoint Analysis with the help of an assessment done on the consumer’s preference towards a particular set of characteristics of a brand or a brand profile.
The researcher working on Conjoint Analysis constructs stimuli that consist of a questionnaire. This questionnaire consists of certain attribute levels of a particular brand under study. This stimulus in Conjoint Analysis is filled-out by the respondents participating in the study.
In order to obtain a valid inference about the study with the help of Conjoint Analysis, it is crucial that the respondents participating in the study respond to the stimulus in an appropriate manner. The respondents in Conjoint Analysis should address the questions of the stimuli according to their desirability.
These evaluations carried out in Conjoint Analysis are reliable only if the subjective evaluations of the respondent are true.
Conjoint Analysis, therefore, addresses various issues. The utilization of Conjoint Analysis is done in order to determine the comparative importance of the crucial characteristics that affect the choice of the consumer. Conjoint Analysis is used in estimating the share of market brands that fluctuates by the level of attributes.
Thus, in a similar manner, Conjoint Analysis can be used by the researcher to assess the consumer’s preference over the attributes of consumer goods, industrial goods, etc. The process of Conjoint Analysis is useful in cases where one needs to address certain issues instead of carrying out the concept of testing. Conjoint Analysis is useful for a person who is not so well versed with statistical skills.
The model that is used by the researcher in Conjoint Analysis to fit the data obtained is the utility function model. This model in Conjoint Analysis is a mathematical model that is used by the researcher to establish a fundamental relationship between the attributes and the utility attached to the product under study.
In Conjoint Analysis, the dependent or the predicted variable is generally the variable that is labeled as the preferences that make the customers attached to a particular brand.
In order to assess the reliability or the validity of Conjoint Analysis, there are several procedures that have been developed.
In Conjoint Analysis, a reliability test called the test retest reliability test, is used by the researcher to obtain identical judgments that are sometimes present in the process of data collection. If the Conjoint Analysis is carried out in a collective manner, then the estimated sample is split into several samples. Then, on each of the split sub samples, Conjoint Analysis is carried out in order to assure whether or not the Conjoint Analysis is valid.
The steps involved while conducting conjoint analysis are the following:
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All of this is determined in Conjoint Analysis with the help of an assessment done on the consumer’s preference towards a particular set of characteristics of a brand or a brand profile.
The researcher working on Conjoint Analysis constructs stimuli that consist of a questionnaire. This questionnaire consists of certain attribute levels of a particular brand under study. This stimulus in Conjoint Analysis is filled-out by the respondents participating in the study.
In order to obtain a valid inference about the study with the help of Conjoint Analysis, it is crucial that the respondents participating in the study respond to the stimulus in an appropriate manner. The respondents in Conjoint Analysis should address the questions of the stimuli according to their desirability.
These evaluations carried out in Conjoint Analysis are reliable only if the subjective evaluations of the respondent are true.
Conjoint Analysis, therefore, addresses various issues. The utilization of Conjoint Analysis is done in order to determine the comparative importance of the crucial characteristics that affect the choice of the consumer. Conjoint Analysis is used in estimating the share of market brands that fluctuates by the level of attributes.
Thus, in a similar manner, Conjoint Analysis can be used by the researcher to assess the consumer’s preference over the attributes of consumer goods, industrial goods, etc. The process of Conjoint Analysis is useful in cases where one needs to address certain issues instead of carrying out the concept of testing. Conjoint Analysis is useful for a person who is not so well versed with statistical skills.
The model that is used by the researcher in Conjoint Analysis to fit the data obtained is the utility function model. This model in Conjoint Analysis is a mathematical model that is used by the researcher to establish a fundamental relationship between the attributes and the utility attached to the product under study.
In Conjoint Analysis, the dependent or the predicted variable is generally the variable that is labeled as the preferences that make the customers attached to a particular brand.
In order to assess the reliability or the validity of Conjoint Analysis, there are several procedures that have been developed.
In Conjoint Analysis, a reliability test called the test retest reliability test, is used by the researcher to obtain identical judgments that are sometimes present in the process of data collection. If the Conjoint Analysis is carried out in a collective manner, then the estimated sample is split into several samples. Then, on each of the split sub samples, Conjoint Analysis is carried out in order to assure whether or not the Conjoint Analysis is valid.
The steps involved while conducting conjoint analysis are the following:
- The first step in conjoint analysis is to form a problem.
- The next step in conjoint analysis is to construct stimuli.
- The third step in conjoint analysis is to choose the form of input data.
- The fourth step of conjoint analysis consists of the selection of the conjoint analysis procedure.
- The fifth step is to infer the results from conjoint analysis.
- The last step is to assess the reliability and validity of conjoint analysis.
Monday, May 25, 2009
Statistical Formula
Statistical formula can be defined as the group of statistical symbols used to make a statistical statement.
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The term called the expected value of some random variable X will be represented by the statistical formula as E(X)= μx=∑[xi * P(xi)]. In this statistical formula, the symbol ‘μx’ represents the expected value of some random variable X. In this statistical formula, the symbol ‘P (xi)’ represents the probability that the random variable will have an outcome ‘i.’ In this statistical formula, the expected value of the random variable X will be computed in the above described manner if the random variable is discreet in nature.
The term called the variance of some random variable X is represented by the statistical formula as Var(X) =σ2 = Σ [Xi – μx]2 * P(xi). In this statistical formula, the symbol ‘σ2’ represents the variance of that random variable.
The term called the chi square statistic will be represented by the statistical formula as X2=[(n-1)*s2]/ σ2. In this statistical formula, the X2 is being represented as the chi square statistic. In this statistical formula, ‘n’ represents the size of the sample. In this statistical formula, ‘s2’ represents the sample variance.
The term called the f statistic will be represented by the statistical formula f=[s12/ σ12]/ [s22/ σ22]. In this statistical formula, s12 represents the variance of the sample drawn from population 1 and s22 represents the variance of the sample drawn from population 2.
The expected value of the sum of two random variables, for example, random variable X and random variable Y, will be represented by the statistical formula as E(X+Y)=E(X)+E(Y). The term E(X) and E(Y) in the statistical formula is nothing but the same as described above.
The expected value of the difference between the random variables will be represented by the statistical formula as E(X-Y) =E(X)-E(Y). The term ‘E(X-Y)’ in the statistical formula is nothing but the expected value of the difference between the random variables.
The variance of the sum of the independent variable is represented by the statistical formula as Var(X+Y) = Var(X)+Var(Y). Ideally, in this statistical formula, the covariance between the two variables should also exist, but since the two variables are independent in nature, the covariance in this statistical formula will not exist.
The standard error of the difference for proportion is represented by the statistical formula as SEp= sp = sqrt [ p*(1-p)*{1/n1 + 1/n2} ] . The term ‘SEp’ in the statistical formula represents the standard error for difference proportion. The term ‘p’ in the statistical formula is the pooled sample variance. The term ‘n1’ in the statistical formula represents the size of the first sample and the term ‘n2’ in the statistical formula represents the size of the second sample, which is pooled with the first sample.
The binomial formula is represented by the statistical formula as P(X=x)=b(x;n,P)= nCx * px(1-p)n-x. The term ‘n’ in this statistical formula represents the number of trials. The term ‘x’ in this statistical formula represents the number of successes in ‘n’ trials. The term ‘p’ in this statistical formula represents the probability of getting success from the ‘n’ binomial trials.
The poisson formula is represented by the statistical formula as P(x;µ)=(e-µ)(µx)/x!. The term ‘µ’ in the statistical formula represents the mean number of the successes that has occurred in a specific region. The term ‘x’ in the statistical formula represents the actual number of successes that has occurred in a specific region. The term ‘e’ in the statistical formula represents the base of the natural logarithmic system. Its value is approximately 2.71828.
For assistance with statistical formula analysis, click here.
The term called the expected value of some random variable X will be represented by the statistical formula as E(X)= μx=∑[xi * P(xi)]. In this statistical formula, the symbol ‘μx’ represents the expected value of some random variable X. In this statistical formula, the symbol ‘P (xi)’ represents the probability that the random variable will have an outcome ‘i.’ In this statistical formula, the expected value of the random variable X will be computed in the above described manner if the random variable is discreet in nature.
The term called the variance of some random variable X is represented by the statistical formula as Var(X) =σ2 = Σ [Xi – μx]2 * P(xi). In this statistical formula, the symbol ‘σ2’ represents the variance of that random variable.
The term called the chi square statistic will be represented by the statistical formula as X2=[(n-1)*s2]/ σ2. In this statistical formula, the X2 is being represented as the chi square statistic. In this statistical formula, ‘n’ represents the size of the sample. In this statistical formula, ‘s2’ represents the sample variance.
The term called the f statistic will be represented by the statistical formula f=[s12/ σ12]/ [s22/ σ22]. In this statistical formula, s12 represents the variance of the sample drawn from population 1 and s22 represents the variance of the sample drawn from population 2.
The expected value of the sum of two random variables, for example, random variable X and random variable Y, will be represented by the statistical formula as E(X+Y)=E(X)+E(Y). The term E(X) and E(Y) in the statistical formula is nothing but the same as described above.
The expected value of the difference between the random variables will be represented by the statistical formula as E(X-Y) =E(X)-E(Y). The term ‘E(X-Y)’ in the statistical formula is nothing but the expected value of the difference between the random variables.
The variance of the sum of the independent variable is represented by the statistical formula as Var(X+Y) = Var(X)+Var(Y). Ideally, in this statistical formula, the covariance between the two variables should also exist, but since the two variables are independent in nature, the covariance in this statistical formula will not exist.
The standard error of the difference for proportion is represented by the statistical formula as SEp= sp = sqrt [ p*(1-p)*{1/n1 + 1/n2} ] . The term ‘SEp’ in the statistical formula represents the standard error for difference proportion. The term ‘p’ in the statistical formula is the pooled sample variance. The term ‘n1’ in the statistical formula represents the size of the first sample and the term ‘n2’ in the statistical formula represents the size of the second sample, which is pooled with the first sample.
The binomial formula is represented by the statistical formula as P(X=x)=b(x;n,P)= nCx * px(1-p)n-x. The term ‘n’ in this statistical formula represents the number of trials. The term ‘x’ in this statistical formula represents the number of successes in ‘n’ trials. The term ‘p’ in this statistical formula represents the probability of getting success from the ‘n’ binomial trials.
The poisson formula is represented by the statistical formula as P(x;µ)=(e-µ)(µx)/x!. The term ‘µ’ in the statistical formula represents the mean number of the successes that has occurred in a specific region. The term ‘x’ in the statistical formula represents the actual number of successes that has occurred in a specific region. The term ‘e’ in the statistical formula represents the base of the natural logarithmic system. Its value is approximately 2.71828.
Canonical Correlation
Canonical correlation was developed by H.Hotelling. Canonical correlation analysis is a method of measuring the linear relationship between two variables that are multidimensional in nature. The canonical correlation analysis locates the bases for each variable that are most favorable with respect to the association. The canonical correlation analysis also determines the corresponding associations. In other words, the canonical correlation analysis determines the two bases in which the association matrix between the variables is diagonal and the associations on the diagonals are maximized. The dimensionality of these new bases in canonical correlation is equal to or less than the smallest dimensionality of the two variables.
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One of the crucial properties of canonical correlation is that they are independent in relation to the transformation of the variables. This property of canonical correlation indicates the difference between the canonical correlation and the ordinary types of correlation.
Canonical correlation is a standard tool in statistical analysis which is used in the fields of economics, medical studies, etc.
In statistical language, the canonical correlation can be defined as the problem of finding two sets of basis vectors in such a manner that associations between the projections of the variables into the basis vectors are mutually maximized.
The canonical correlation between the two random vectors can be obtained by calculating the Eigen value equations. The Eigen values are nothing, but are equivalent to the square of the canonical correlation.
The canonical correlation is a member of the multiple general linear hypothesis family and contributes most of the assumptions of multiple regression, such as the linearity of relationships, homoscedasticity, interval level of data, proper specification of the model, lack of high multicollinearity, etc. The canonical correlation is also called a characteristic root.
The maximum number of canonical correlation between the two sets of variables is the number of variables in the smaller set.
The pooled canonical correlation is the sum of squares of all the canonical coefficients and it represents all the orthogonal dimensions in the solution by which the two sets of variables are associated. The pooled canonical correlation is used to extract the extent to which one set of variables can be forecasted by the other set of variables.
The canonical weights are nothing but the canonical coefficient in canonical correlation, which is used to assess the comparative importance that is contributed by the individual variable to a given canonical correlation.
The canonical scores in the canonical correlation are the values that are assigned to the canonical variable for a particular case. This score of canonical correlation is based on the value of the canonical coefficients for that variable. The canonical coefficients in canonical correlation are multiplied by the scores that are standardized and are then summed to yield canonical scores.
The structure correlation coefficients in canonical correlation are also called canonical factor loadings. It is defined as the canonical correlation of a canonical variable with an original variable in its set. The squared structure correlations in canonical correlation depict the contribution of a variable to the explanatory power of the canonical variate based on the set of variables.
For assistance with canonical correlation analysis, click here.
One of the crucial properties of canonical correlation is that they are independent in relation to the transformation of the variables. This property of canonical correlation indicates the difference between the canonical correlation and the ordinary types of correlation.
Canonical correlation is a standard tool in statistical analysis which is used in the fields of economics, medical studies, etc.
In statistical language, the canonical correlation can be defined as the problem of finding two sets of basis vectors in such a manner that associations between the projections of the variables into the basis vectors are mutually maximized.
The canonical correlation between the two random vectors can be obtained by calculating the Eigen value equations. The Eigen values are nothing, but are equivalent to the square of the canonical correlation.
The canonical correlation is a member of the multiple general linear hypothesis family and contributes most of the assumptions of multiple regression, such as the linearity of relationships, homoscedasticity, interval level of data, proper specification of the model, lack of high multicollinearity, etc. The canonical correlation is also called a characteristic root.
The maximum number of canonical correlation between the two sets of variables is the number of variables in the smaller set.
The pooled canonical correlation is the sum of squares of all the canonical coefficients and it represents all the orthogonal dimensions in the solution by which the two sets of variables are associated. The pooled canonical correlation is used to extract the extent to which one set of variables can be forecasted by the other set of variables.
The canonical weights are nothing but the canonical coefficient in canonical correlation, which is used to assess the comparative importance that is contributed by the individual variable to a given canonical correlation.
The canonical scores in the canonical correlation are the values that are assigned to the canonical variable for a particular case. This score of canonical correlation is based on the value of the canonical coefficients for that variable. The canonical coefficients in canonical correlation are multiplied by the scores that are standardized and are then summed to yield canonical scores.
The structure correlation coefficients in canonical correlation are also called canonical factor loadings. It is defined as the canonical correlation of a canonical variable with an original variable in its set. The squared structure correlations in canonical correlation depict the contribution of a variable to the explanatory power of the canonical variate based on the set of variables.
Multicollinearity
Multicollinearity depicts the stage in which the independent variables in the data show high interactions and associations. Therefore, multicollinearity is considered a disturbance that causes volatility in data.
Contact Statistics Solutions today for assistance with identifying multicollinearity in data.
There are reasons behind the outcome of multicollinearity in data.
Multicollinearity can occur due to the improper utilization of dummy variables. Researchers who are not experts can end up causing multicollinearity in the data.
If the researcher includes a variable which is being computed from the other variables in the equation, then this action can cause multicollinearity in the data. For example, if the family’s health is equal to the husband’s health+ wife’s health + child’s health, and the regression includes all four health cases, then this calls for multicollinearity.
If a researcher includes the same type of variables twice in an experiment, then this activity performed by the researcher causes multicollinearity. For example, if two models of Nokia phones are included as different variables in the study, then this causes multicollinearity in the data.
There are certain outcomes of multicollinearity.
The researcher should note that as the level of multicollinearity increases, the value of the standard error gets higher and higher. When there is high multicollinearity in the data, then the confidence intervals for the coefficients tend to be extremely wide and the value of the t-statistics tends to be very small. The researcher should keep in mind that the value of the coefficients should be larger in order to have it statistically significant. In other words, in the presence of multicollinearity, the null hypothesis assumed by the researcher is harder to get rejected.
If the value of the tolerance is closer to the value of one, then this means that there is very little multicollinearity. On the other hand, if the value of the tolerance is closer to zero then this means that there is very high multicollinearity. So, in the latter case, the multicollinearity is considered a threat.
The reciprocal of the tolerance is known as the variance inflation factor (VIF). The variance inflation factor shows the amount of variance of the coefficient estimate that is inflated by multicollinearity.
Multicollinearity is not something that is discreet in nature, but is a matter of degree. The matter of multicollinearity can be detected with the help of certain warning signals.
If the t ratios for each coefficient are statistically significant, and the F statistic is not statistically significant, then this indicates that there is multicollinearity in the data.
It is important to check the stability of the coefficients when two different types of samples are used. If the coefficients differ quiet significantly, then this shows that there is multicollinearity in the data.
If the sign of the variables gets changed, or if some variables are being added, then this signifies the presence of multicollinearity.
In order to address the problem of multicollinearity, one has to make sure that he/she does not conduct improper usage of the dummy variables.
If the sample size is increased from the desired sample size, then this will decrease the value of the standard errors and simultaneously it would decrease the level of multicollinearity.
It is sometimes suggested that the researcher drop the variable that is causing multicollinearity. The researcher should keep in mind that if the most important variable is dropped by the researcher, then this would cause a specification error, which is even worse than multicollinearity.
The most important thing for obtaining a valid inference about the data is to realize the presence of multicollinearity. Additionally, a researcher should be aware of the consequences of multicollinearity.
Contact Statistics Solutions today for assistance with identifying multicollinearity in data.
There are reasons behind the outcome of multicollinearity in data.
Multicollinearity can occur due to the improper utilization of dummy variables. Researchers who are not experts can end up causing multicollinearity in the data.
If the researcher includes a variable which is being computed from the other variables in the equation, then this action can cause multicollinearity in the data. For example, if the family’s health is equal to the husband’s health+ wife’s health + child’s health, and the regression includes all four health cases, then this calls for multicollinearity.
If a researcher includes the same type of variables twice in an experiment, then this activity performed by the researcher causes multicollinearity. For example, if two models of Nokia phones are included as different variables in the study, then this causes multicollinearity in the data.
There are certain outcomes of multicollinearity.
The researcher should note that as the level of multicollinearity increases, the value of the standard error gets higher and higher. When there is high multicollinearity in the data, then the confidence intervals for the coefficients tend to be extremely wide and the value of the t-statistics tends to be very small. The researcher should keep in mind that the value of the coefficients should be larger in order to have it statistically significant. In other words, in the presence of multicollinearity, the null hypothesis assumed by the researcher is harder to get rejected.
If the value of the tolerance is closer to the value of one, then this means that there is very little multicollinearity. On the other hand, if the value of the tolerance is closer to zero then this means that there is very high multicollinearity. So, in the latter case, the multicollinearity is considered a threat.
The reciprocal of the tolerance is known as the variance inflation factor (VIF). The variance inflation factor shows the amount of variance of the coefficient estimate that is inflated by multicollinearity.
Multicollinearity is not something that is discreet in nature, but is a matter of degree. The matter of multicollinearity can be detected with the help of certain warning signals.
If the t ratios for each coefficient are statistically significant, and the F statistic is not statistically significant, then this indicates that there is multicollinearity in the data.
It is important to check the stability of the coefficients when two different types of samples are used. If the coefficients differ quiet significantly, then this shows that there is multicollinearity in the data.
If the sign of the variables gets changed, or if some variables are being added, then this signifies the presence of multicollinearity.
In order to address the problem of multicollinearity, one has to make sure that he/she does not conduct improper usage of the dummy variables.
If the sample size is increased from the desired sample size, then this will decrease the value of the standard errors and simultaneously it would decrease the level of multicollinearity.
It is sometimes suggested that the researcher drop the variable that is causing multicollinearity. The researcher should keep in mind that if the most important variable is dropped by the researcher, then this would cause a specification error, which is even worse than multicollinearity.
The most important thing for obtaining a valid inference about the data is to realize the presence of multicollinearity. Additionally, a researcher should be aware of the consequences of multicollinearity.
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