## Using Multivariate Statistics (Tabachnik & Fidell, 6th edition)

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Summary of Chapter 3. Univariate and bivariate statistics

Book: Using Multivariate Statistics (Tabachnick & Fidell, 6th edition)

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Using Multivariate Statistics (Tabachnik & Fidell, 6th edition)

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What are multivariate statistics and how to use them? - Chapter 1

- How does multivariate statistics differ from univariate and bivariate statistics?
- What is the difference between experimental and nonexperimental research?
- Why is multivariate statistics gaining popularity?
- What are common terms in multivariate statistics?
- How does multivariate statistics combine variables?
- How many variables should you include in your analysis?
- How to control for adequate power?
- What data are appropriate for multivariate statistics?

Multivariate statistics consists of a range of statistical techniques that deal with the analysis and interpretation of data sets containing multiple variables simultaneously. Multivariate statistics provides analysis when there are many independent variables (IVs) and/or many dependent variables (DVs), all correlated with one another to some degree. Unlike univariate statistics, which focus on analyzing one variable at a time, multivariate statistics examines the relationships and interactions among multiple variables in a dataset. It is widely used in various disciplines, including psychology, economics, biology, sociology, marketing, and more. The main goal of multivariate statistics is to uncover patterns, associations, and dependencies among the variables in a dataset to gain a deeper understanding of the underlying structure and complexity of the data.

First, it is important to distinguish between dependent and independent variables. *Independent variables* (IVs) are the differing conditions (treatment vs. placebo) to which you expose your subjects, or the characteristics (tall or short) that the subjects themselves bring into the research experiment. IVs are usually considered *predictor* variables because they predict the *dependent variables*, also called the response or outcome variables. Note that independent and dependent variables are defined within a research context; a DV in one research setting may be an IV in another.

Multivariate statistics, univariate statistics, and bivariate statistics are different approaches to analyzing data based on the number of variables involved in the analysis:

*Univariate Statistics*: Univariate statistics deal with the analysis of a single dependent variable at a time. There may be, however, more than one IV. The main goal of univariate analysis is to describe and summarize the distribution, central tendency, and dispersion of a single variable. Common univariate techniques include calculating measures such as mean, median, mode, standard deviation, variance, and conducting graphical representations like histograms and box plots. Univariate analysis provides basic insights into the characteristics of individual variables but does not consider the relationships between variables.*Bivariate Statistics*: Bivariate statistics involve the analysis of two variables simultaneously, where neither is an experimental IV. The primary focus of bivariate analysis is to explore the relationship between two variables. Common bivariate techniques include correlation analysis, which measures the strength and direction of the linear relationship between two continuous variables, and cross-tabulation or contingency tables, used for categorical variables. Bivariate statistics are useful

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How to organize statistical techniques? - Chapter 2

This chapter explains how to organize the statistical techniques in this book by major research question. In summary, a decision tree leads you to an appropriate analysis for your data. On the basis of your major research question and a few characteristics of your data set, you determine which statistical technique(s) is appropriate.

To determine which analytic strategy to use, you should consider the following steps:

- Major research question
- Number (and kind) of dependent variables
- Number (and kind) of independent variables
- Covariates

Hence, the analytic strategy is to be determined based on the major research question, the number (and kind) of dependent and independent variables and the presence (or absence) of covariates. This chapter briefly introduces the statistical techniques. However, the focus is on when to choose which technique rather than to discuss each technique into detail. To do so, each paragraph describes the techniques belonging to one of the following major research questions:

- Degree of relationship among variables
- Significance of group differences
- Prediction of group membership
- Structure
- Time course of events

If the major purpose of analysis is to assess the associations among two or more variables, there are five statistical techniques you could potentially use. The choice among these five different statistical techniques depends on the number of independent and dependent variables, the nature of the variables (continuous or discrete), and whether any of the independent variables is best conceptualized as covariate.

Major research question | Number (and kind) of DVs | Number (and kind) of IVS | Covariates | Analytic strategy | Goal of analysis |

Degree of relationship among variables | One (continuous) | One (continuous) | - |
(see Chapter 17) | Create a linear combination of IVs to optimally predict DV. |

,, | ,, | Multiple (continuous) | None |
(see Chapter 5) | Create a linear combination of IVs to optimally predict DV. |

,, | ,, | ,, | Some |
(see Chapter 5) | Create a linear combination of IVs to optimally predict DV. |

,, | Multiple (continuous) | Multiple (continuous) | - |
(see Chapter 12) | Maximally correlate a linear combination of DVs with a linear combination of IVs. |

,, | One (may be repeated) | Multiple (continuous and discrete; cases and IVs are nested) | - |
(see Chapter 16) | Create linear combinations of IVs at one level to serve as DVs at another level. |

,, | None | Multiple (discrete) | - |
(see Chapter 15) | Create a log-linear combination of IVs to optimally predict category frequencies. |

In studies where participants are randomly assigned to different groups (treatments), the main focus is usually on determining if there are

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What are the core concepts of univariate and bivariate statistics? - Chapter 3

Statistics are essential for making informed decisions when dealing with uncertain situations. Inferences or decisions are drawn about entire populations based on data collected from smaller samples, which may not fully represent the population. As a result, any conclusions about the population carry some degree of risk. To address this issue, statistical decision theory is commonly used. It involves formulating two hypothetical scenarios, each represented by a probability distribution, as alternative explanations for the data. Based on the observed sample results, formal statistical rules are applied to determine the most likely scenario, which serves as the best estimate for the underlying truth of events. However, it is essential to recognize that uncertainty still exists, and any conclusions drawn are subject to some level of uncertainty or risk.

The *one-sample z-tes*t is a statistical test used to compare the mean of a single sample to a known population mean or a hypothesized value. It assesses whether there is a significant difference between the sample mean and the population mean, helping researchers draw conclusions about the population based on the observed sample data. The test relies on the standard normal distribution and assumes that the sample is drawn from a normally distributed population or has a sufficiently large sample size for the central limit theorem to apply.

Note that, in hypothesis testing, we examine means rather than individual scores. Hypothetical scenarios represent distributions of means, not individual scores. These "sampling distributions of means" differ from distributions of individual scores in a systematic manner. The focus is on comparing and drawing conclusions about mean values rather than individual data points. Sampling distributions have smaller standard deviations than distributions of scores, and the decrease is related to N, the sample size.

H_{0} (null hypothesis) and H_{a }(alternative hypothesis) represent two possible scenarios, only one of which is true. When researchers have to decide whether to accept or reject H_{0}, four possible outcomes can occur. If H_{0} is true, a correct decision is made when retaining H_{0}, and an error is made when rejecting it. The probability of making the wrong decision (Type I error) is denoted as "a," while the probability of making the correct decision is "1 - α." On the other hand, if H_{a} is true, a correct decision is made when rejecting H_{0}, and an error occurs when retaining it. The probability of making the wrong decision (Type II error) is denoted as "β," and the probability of making the correct decision is "1

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How to clean up a data set? - Chapter 4

This chapter focuses on data preparation and resolution of various issues before the analysis to ensure an honest and accurate data analysis, which may be time-consuming but is crucial for the integrity of the results. The topics covered in this chapter include:

- Accuracy of data entry and considerations for avoiding distorted correlations.
- Dealing with missing data.
- Assessing and ensuring the fit between data and assumptions of multivariate procedures.
- Transformations of variables to meet analysis requirements.
- Handling outliers (extreme cases) that can influence and distort solutions.
- Addressing perfect or near-perfect correlations among variables that may pose challenges in multivariate analysis.

To ensure data accuracy in small data files, proofreading the original data against the computerized data file is recommended, while for large data sets, screening involves examining univariate descriptive statistics and graphic representations, checking for plausible values, and accurate coding for missing values.

Most multivariate procedures analyze patterns of correlation (or covariance) among variables. It is important that the correlations, whether between two continuous variables or between a dichotomous and continuous variable, be as accurate as possible. Under some rather common research conditions, correlations are larger or smaller than they should be.

*Inflated correlation* refers to a correlation between variables that appears higher than it should be due to the presence of shared items or overlapping components in the measurement of those variables. In social and behavioral sciences, variables are often composites of several items. If composite variables are used and they contain, in part, the same items, correlations are inflated; do not overinterpret a high correlation between two measures composed, in part, of the same items. If there is enough overlap, consider using only one of the composite variables in the analysis.

*Deflated correlation* refers to a correlation between variables that appears lower than it should be due to the absence or exclusion of shared items or overlapping components in the measurement of those variables. A falsely small correlation between two continuous variables is obtained if the range of responses to one or both of the variables is restricted in the sample. When a correlation is too small because of restricted range in sampling, you can estimate its magnitude in a nonrestricted sample.

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What is multiple regression? - Chapter 5

- What is the standard form of the equation for multiple regression?
- What is the aim of multiple regression?
- What kind of research questions can be answered by multiple regression??
- What are limitations of mulitple regression?
- What are the three major types of multiple regression?
- How does multiple regression differ from ANOVA?
- Example

*Multiple regression* is a statistical technique used to analyze the relationship between a dependent variable and two or more independent variables. It extends simple linear regression to account for multiple predictors simultaneously. The method estimates the unique contribution of each independent variable to the dependent variable while controlling for the effects of other predictors. Multiple regression is widely employed in various fields to model and predict complex relationships between variables and is useful for hypothesis testing, prediction, and understanding the influence of multiple factors on an outcome.

The standard form of the equation for multiple regression with *K* independent variables (predictors) can be represented as follows:

Y' = A + B_{1}X_{1 }+ B_{2}X_{2 } + ... + B_{K}X_{K}

Where:

- Y' is the predicted value on the DV
- A is the Y intercept (the value of Y when all the X values are zero)
- the Xs represent the various IVs (of which there are K)
- the Bs are the regression coefficients assigned to each of the IVs during regression

The best-fitting regression coefficients produce a prediction equation for which squared differences between Y and Y' are at a minimum. Because squared errors of prediction, (Y - Y) 2 , are minimized, this solution is called a least- squares solution.

The aim of multiple regression is to model and analyze the relationship between a dependent variable (response variable) and two or more independent variables (predictor variables). The main objective is to understand how the independent variables jointly influence the dependent variable and to estimate the regression coefficients (weights) that represent the strength and direction of these relationships. This enables us to make predictions, understand the relative importance of different predictors, and identify which variables have a significant impact on the outcome of interest. Multiple regression is a fundamental tool for analyzing complex relationships.

Multiple regression can be used to answer a wide range of research questions. Some examples of research questions that can be addressed using multiple regression include:

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What is analysis of covariance (ANCOVA)? – Chapter 6

Analysis of Covariance (ANCOVA) is a statistical technique that combines elements of both analysis of variance (ANOVA) and regression analysis. It is used to compare the means of two or more groups while controlling for the influence of one or more continuous covariates (also known as independent variables or predictors) on the outcome variable. In ANCOVA, the covariates are continuous variables that could potentially influence the outcome variable. ANCOVA extends the traditional ANOVA by accounting for the covariates' influence, thus allowing researchers to isolate the unique effect of the grouping variable on the outcome variable while controlling for potential confounding variables. This helps in reducing variability and increasing the statistical power to detect group differences accurately. The primary goal of ANCOVA is to examine whether there are significant differences between the group means on the outcome variable after adjusting for the effects of the covariates.

ANCOVA is a versatile statistical technique that can be used to answer various research questions in different fields. Some common research questions that can be addressed using ANCOVA include:

- Treatment comparison: Is there a significant difference in the outcome variable between two or more groups (e.g., experimental conditions, treatment groups) after controlling for the influence of one or more covariates?
- Intervention effectiveness: Does a specific intervention or treatment have a significant effect on the outcome variable while accounting for the influence of other variables, such as pre-existing differences or baseline scores?
- Group comparisons with covariates: Are there significant group differences on the outcome variable when considering the impact of certain covariates (e.g., age, gender, socioeconomic status) that may influence the outcome?
- Analysis of experimental data: In experimental research, ANCOVA can be used to determine if the treatment effect remains significant after controlling for potential confounding variables.
- Analysis of survey data: In survey research, ANCOVA can help explore whether differences in survey responses across groups are significant when accounting for other relevant factors.
- Comparative Studies: ANCOVA can be used to compare the performance or outcomes of different groups (e.g., students from different schools or regions) while adjusting for relevant covariates.
- Longitudinal studies: In longitudinal studies, ANCOVA can assess changes in outcomes over time between groups while accounting for covariate effects.
- Controlling for extraneous variables: ANCOVA is valuable when researchers want to control for potential confounding variables to isolate the effects of the independent variable of interest.

Overall, ANCOVA is a valuable tool for researchers who wish to investigate group differences or the impact of interventions while taking into account the influence of

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When and how to use MANOVA and MANCOVA? – Chapter 7

MANOVA and MANCOVA are statistical techniques for analyzing multiple dependent variables in relation to one or more independent variables. MANOVA examines how the independent variables affect the dependent variables simultaneously, providing efficiency and multivariate insights. On the other hand, MANCOVA extends MANOVA by including continuous covariates, offering the advantage of controlling for confounding variables and increasing precision in estimating the effects of independent variables on dependent variables. However, both methods have assumptions to consider, and MANCOVA may require a larger sample size.

*Multivariate Analysis of Variance (MANOVA)* is a statistical technique used to analyze the relationships among several dependent variables simultaneously when they are influenced by one or more independent variables. It extends the concept of univariate Analysis of Variance (ANOVA) to situations where there are multiple dependent variables. MANOVA is particularly useful when you want to understand how different independent variables impact multiple related dependent variables.

The advantages of using MANOVA include its efficiency, as it can provide more powerful results compared to conducting separate ANOVA tests for each dependent variable. By analyzing multiple dependent variables together, MANOVA offers a more comprehensive understanding of their interrelationships and how they respond to changes in the independent variables. Additionally, the statistical power of MANOVA tends to be higher than conducting univariate analyses.

However, MANOVA has some disadvantages. One significant assumption is that the data should follow a multivariate normal distribution, and the variance-covariance matrices should be homogeneous across groups. Violating these assumptions can lead to biased results. Moreover, interpreting MANOVA results can be more complex than interpreting ANOVA results due to the multivariate nature of the analysis. Lastly, MANOVA typically requires a larger sample size, especially when there are multiple dependent variables.

*Multivariate Analysis of Covariance (MANCOVA)* is a technique that extends MANOVA by incorporating one or more continuous covariates (control variables) in addition to the independent variables. Including covariates allows researchers to control for the influence of these continuous variables while examining the effects of the independent variables on multiple dependent variables.

The primary advantage of MANCOVA is that it enables researchers to control for confounding variables. By including covariates in the analysis, it becomes possible to isolate the effects of the independent variables on the dependent variables more accurately. This increased precision can be beneficial in drawing more accurate conclusions from the data.

However, just like MANOVA, MANCOVA has its own set of assumptions and limitations. The data still need to satisfy the assumptions of multivariate normality, homogeneity of variance-covariance matrices, and independence of observations. Violations of these assumptions can compromise the validity of the results. Moreover, including covariates in the analysis can add complexity to result interpretation. Additionally, using MANCOVA often requires a larger sample size to produce reliable and meaningful outcomes.

Multivariate Analysis of Variance (MANOVA) and Multivariate Analysis of Covariance (MANCOVA) are statistical methods used to answer research questions that involve multiple dependent variables (DV) and one or more

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What is profile analysis? - Chapter 8

Profile analysis is a multivariate statistical technique used to analyze repeated measures data with multiple dependent variables. In a repeated measures design, the same participants are measured on multiple occasions or under different conditions. Profile analysis allows researchers to examine the patterns of change or differences across the dependent variables over time or across conditions. The main goal of profile analysis is to test the significance of the interaction between the repeated measures factor (time or condition) and the dependent variables. If the interaction is significant, it indicates that the effect of the independent variable (time or condition) varies across the dependent variables.

Profile analysis is particularly suited for research questions that involve the examination of changes or differences in multiple dependent variables across repeated measures or conditions. Some specific research questions that can be answered using profile analysis include:

- How do individuals' scores on multiple psychological constructs change over time in a longitudinal study?
- Are there differences in cognitive abilities across different age groups in a cross-sectional study?
- Does the effectiveness of various teaching methods differ for different subject areas in an educational intervention study?
- How do participants' emotional responses vary across different experimental conditions in a psychological experiment?
- Are there differences in physical performance across different training programs in a sports science study?

In all these research questions, the focus is on examining the patterns of change or differences in multiple dependent variables across repeated measures or conditions. Profile analysis allows researchers to gain a comprehensive understanding of the interrelationships and interactions among the dependent variables, providing valuable insights into the underlying processes and relationships being studied.

Profile analysis, like any statistical method, has its limitations and considerations that researchers should be aware of:

- Sample size: Profile analysis requires a sufficient sample size to yield reliable results, especially when dealing with multiple dependent variables and conditions. Small sample sizes can lead to less stable estimates and reduced statistical power.
- Assumption of multivariate normality: Profile analysis assumes that the data follow a multivariate normal distribution. If the assumption is violated, the results may not be accurate or valid. Researchers should check for normality and consider data transformations if necessary.
- Homogeneity of covariance matrices: Profile analysis assumes that the covariance

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What is discriminant analysis? - Chapter 9

- What kind of research questions can be answered with discriminant analysis?
- What are limitations to discriminant analysis?
- What is the basic equation of discriminant analysis?
- What are the three types of discriminant analysis?
- What are important issues to consider when using discriminant analysis?
- How to interpret discriminant functions?
- How to use discriminant analysis: An example

Discriminant analysis is a statistical method used to classify objects or cases into predefined groups based on a set of predictor variables. The goal of discriminant analysis is to find a linear combination of the predictor variables that maximally separates the groups and minimizes the within-group variability. It is commonly employed in situations where there are two or more groups, and the objective is to determine which combination of variables can best discriminate between them. Discriminant analysis is widely used in various fields, such as psychology, medicine, and marketing, to predict group membership or to identify the key variables that contribute to group differences.

Discriminant analysis can be used to address a variety of research questions related to group classification and prediction. Some common research questions that can be answered using discriminant analysis include:

- Group classification: Discriminant analysis can be used to classify individuals or cases into predefined groups based on their characteristics or predictor variables. For example, it can be used to classify patients into different disease categories based on their symptoms or to distinguish between different customer segments based on their purchasing behavior.
- Group differences: Discriminant analysis can identify the key variables that contribute to group differences. Researchers can use discriminant analysis to determine which variables are most important in distinguishing between groups and to understand how these variables differ across groups.
- Predictive modeling: Discriminant analysis can also be used for prediction. Researchers can build a discriminant model using a set of predictor variables to predict group membership or outcomes for new cases. For instance, it can be used to predict whether a student will succeed or struggle in a particular academic program based on their demographic and academic characteristics.
- Variable selection: Discriminant analysis can help researchers identify the most relevant variables for group classification. By examining the variable weights or contributions in the discriminant function, researchers can determine which variables are most influential in distinguishing between groups.

Overall, discriminant analysis is a powerful tool for understanding group differences, predicting group membership, and selecting important variables in research studies involving multiple groups or categories.

Discriminant analysis has several limitations that researchers should be aware of:

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What is logistic regression? - Chapter 10

- What kind of research questions can be answered with logistic regression?
- What are limitations of logistic regression?
- What is the fundamental equation for logistic regression?
- What are the three types of logistic regression?
- What is probit analysis and how is it related to logistic regression?
- What are important issues to consider when using logistic regression?
- How to use logistic regression: An example

Logistic regression is a statistical method used to model and analyze the relationship between a binary (dichotomous) dependent variable and one or more independent variables (predictors). It is particularly suitable for situations where the dependent variable takes one of two possible outcomes, such as "yes" or "no," "success" or "failure," "1" or "0."

The goal of logistic regression is to estimate the probability that the dependent variable belongs to a particular category based on the values of the independent variables. It uses the logistic function (also known as the sigmoid function) to model the relationship between the predictors and the probability of the event occurring. The output of logistic regression is the predicted probability of the event, which is then used to make a binary decision or classification.

In contrast to linear regression, which predicts continuous outcomes, logistic regression deals with discrete outcomes and is commonly used in various fields, such as medicine, social sciences, marketing, and machine learning, for tasks like predicting whether a customer will purchase a product, diagnosing diseases, or analyzing the likelihood of success in a given situation.

Logistic regression is well-suited for research questions involving binary outcomes or categorical data with two categories. It can answer questions related to the probability or likelihood of an event happening based on predictor variables. Some examples of research questions that can be addressed with logistic regression are:

- Predictive Modeling: Will a customer buy a product based on their demographic characteristics and past purchase behavior?
- Medical Diagnosis: Does a patient have a specific disease based on their symptoms and medical history?
- Educational Outcomes: What factors influence the likelihood of a student passing an exam or graduating from a program?
- Social Sciences: How do certain factors affect the probability of a person voting in an election or participating in a social activity?
- Behavior Analysis: What factors are associated with the likelihood of engaging in a particular behavior, such as smoking or exercise?

In general, logistic regression is used when the outcome of interest is binary or can be dichotomized, and the researcher wants to understand how different independent variables or predictors influence the likelihood of that outcome occurring. It is a valuable tool for understanding relationships between categorical variables and making predictions in situations where

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