Mediation analysis practices in social and personality psychology would benefit from the integration of practices from statistical mediation analysis, which is currently commonly implemented in social and personality psychology, and causal mediation analysis, which is not frequently used in psychology. In this chapter, I briefly describe each method on its own, then provide recommendations for how to integrate practices from each method to simultaneously evaluate statistical inference and causal inference as part of a single analysis. At the end of the chapter, I describe additional areas of recent development in mediation analysis that that social and personality psychologists should also consider adopting I order to improve the quality of inference in their mediation analysis: latent variables and longitudinal models. Ultimately, this chapter is meant to be a kind introduction to causal inference in the context of mediation with very practical recommendations for how one can implement these practices in one’s own research.

This chapter is devoted to extensive instruction regarding bivariate regression, also known as ordinary least squares regression (OLS).Students are presented with a scatterplot of data with a best-fitting line drawn through it.They are instructed on how to calculate the equation of this line (least squares line) by hand and with the R Commander.Interpretation of the statistical output of the y-intercept, beta coefficient, and R-squared value are discussed.Statistical significance of the beta coefficient and its implications for the relationship between an independent and dependent variable are described.Finally, the use of the regression equation for prediction is illustrated.

Many applications require solving a system of linear equations 𝑨𝒙 = 𝒚 for 𝒙 given 𝑨 and 𝒚. In practice, often there is no exact solution for 𝒙, so one seeks an approximate solution. This chapter focuses on least-squares formulations of this type of problem. It briefly reviews the 𝑨𝒙 = 𝒚 case and then motivates the more general 𝑨𝒙 ≈ 𝒚 cases. It then focuses on the over-determined case where 𝑨 is tall, emphasizing the insights offered by the SVD of 𝑨. It introduces the pseudoinverse, which is especially important for the under-determined case where 𝑨 is wide. It describes alternative approaches for the under-determined case such as Tikhonov regularization. It introduces frames, a generalization of unitary matrices. It uses the SVD analysis of this chapter to describe projection onto a subspace, completing the subspace-based classification ideas introduced in the previous chapter, and also introduces a least-squares approach to binary classifier design. It introduces recursive least-squares methods that are important for streaming data.

In this chapter we cover clustering and regression, looking at two traditional machine learning methods: k-means and linear regression. We briefly discuss how to implement these methods in a non-distributed manner first, to then carefully analyze the bottlenecks of these methods when manipulating big data. This enables us to design global-based solutions based on the DataFrame API of Spark. The key focus is on the principles for designing solutions effectively. Nevertheless, some of the challenges in this chapter are to investigate tools from Spark to speed up the processing even further. k-means is an example of an iterative algorithm, and how to exploit caching in Spark, and we analyze its implementation with both RDD and DataFrame APIs. For linear regression, we first implement the closed form, which involves numerous matrix multiplications and outer products, to simplify the processing in big data. Then, we look at gradient descent. These examples give us the opportunity to expand on the principles of designing a global solution, and also allow us to show how knowing the underlying platform, Spark in this case, well is essential to really maximize the performance.

In this chapter, we introduce some of the more popular ML algorithms. Our objective is to provide the basic concepts and main ideas, how to utilize these algorithms using Matlab, and offer some examples. In particular, we discuss essential concepts in feature engineering and how to apply them in Matlab. Support vector machines (SVM), K-nearest neighbor (KNN), linear regression, Naïve Bayes algorithm, and decision trees are introduced and the fundamental underlying mathematics is explained while using Matlab’s corresponding Apps to implement each of these algorithms. A special section on reinforcement learning is included, detailing the key concepts and basic mechanism of this third ML category. In particular, we showcase how to implement reinforcement learning in Matlab as well as make use of some of the Python libraries available online and show how to use reinforcement learning for controller design.

Simple linear regression is extended to multiple linear regression (for multiple predictor variables) and to multivariate linear regression for (multiple response variables). Regression with circular data and/or categorical data is covered. How to select predictors and how to avoid overfitting with techniques such as ridge regression and lasso are followed by quantile regression. The assumption of Gaussian noise or residual is removed in generalized least squares, with applications to optimal fingerprinting in climate change.

Analysis of various data sets can be accomplished using techniques based on least-squares methods.For example, linear regression of data determines the best-fit line to the data via a least-squares approach.The same is true for polynomial and regression methods using other basis functions.Curve fitting is used to determine the best-fit line or curve to a particular set of data, while interpolation is used to determine a curve that passes through all of the data points.Polynomial and spline interpolation are discussed.State estimation is covered using techniques based on least-squares methods.

Research on gender differences in language use previously focused mainly on affluent, especially Western societies. The present chapter extends this research to acrolectal Indian English, a postcolonial variety of English, investigating how the use of intensifiers (e.g. very, really) is affected not only by the speakers’ gender, but also their age, the gender of the other speakers in the conversation and the formality of the context. Results show some parallels with Western varieties of English, in particular a tendency for women to use more intensifiers than men in informal contexts. However, Indian women modify their usage of intensifiers with respect to the formality of the context more than British women and men, while Indian men do so less than British women and men. In mixed-sex conversations, Indian women also converge with Indian men in their intensifier usage, while neither British women nor men do so. The more flexible use of intensifiers by Indian women may be a response to societal expectations regarding their linguistic behaviour, in order to avoid censure by society. British women likewise continue to be affected by such constraints, but much less so, while the linguistic behaviour of Indian and British men is subject to less criticism.

This chapter is not about one particular method (or a family of methods). Instead, it provides a set of tools useful for better pattern recognition, especially for real-world applications. They include the definition of distance metrics, vector norms, a brief introduction to the idea of distance metric learning, and power mean kernels (which is a family of useful metrics). We also establish by examples that proper normalizations of our data are essential, and introduce a few data normalization and transformation methods.

Tognini-Bonelli (2001) made the following distinction between corpus-based and corpus-driven studies. While corpus-based studies start with pre-existing theories which are tested using corpus data, in corpus driven studies the hypothesis is derived by examination of the corpus evidence. This chapter will give an overview of the two different families of statistical tests which are suited for these two approaches. For corpus-based approaches, we use more traditional statistics, such as the t-test, or ANOVA which return a value called a p-value to tell us to what extent we should accept or reject the initial hypothesis. Multi-level modelling (also known as mixed modelling) is a new technique which shows considerable promise for corpus-based studies, and will also be described here to analyse the ENNTT subset of Europarl corpus. Multi-level modelling is useful for the examination of hierarchically structured or “nested” data, where for example translations may be “nested” together in a class if they have the same language of origin. A multi-level model takes account both of the variation between individual translations and the variation between classes. For example, we might expect the scores (such as vocabulary richness or readability scores) of two translations in the same class to be more similar to each other than two translations in different classes.