describing relationships between variables in research

Researchers carefully analyze and interpret the value (s) of each variable to make sense of how things relate to each other in a descriptive study or what has happened in an experiment. Were washing out the part of the \(X\)/\(Y\) relationship that is explained by \(Z\). This brings us to the concept of correlation. What is the relationship between students' achievement motivation and GPA? A correlation coefficient is a bivariate statistic when it summarizes the relationship between two variables, and it's a multivariate statistic when you have more than two variables. The mean was 45. If its negative, theyre negatively related. An official website of the United States government. The fifth column lists the cross-products. In this section, we revisit the two basic forms of statistical relationship introduced earlier in the bookdifferences between groups or conditions and relationships between quantitative variablesand we consider how to describe them in more detail. In general, line graphs are used when the variable on thex-axis has (or is organized into) a small number of distinct values, such as the four quartiles of the name distribution. \[\begin{equation} Table 4.1: Proportion Taking Vitamin E by Range of Body Mass Index Values. So instead of. Because restriction of range is not always anticipated or easily avoidable, however, it is good practice to examine your data for possible restriction of range and to interpret Pearsonsrin light of it. . The main statistical analysis methods were as follows: (1) univariate descriptive statistics were used to analyze the current status of higher vocational students' demographic information, teacher support, procrastination behavior, interpersonal assistance and positive emotions; (2) an independent samples t . A variable can collect either qualitative or quantitative data. But it turns out theres a little magic in there. Figure 4.1: Age and Heart Health, 150 Observations. and transmitted securely. 3) the cause must precede the effect in time. Confounding variables are extra variables, which can have an effect on the experiment. We will first go through the sample size calculation for a hypothesis-based design (like a randomized control trial). Differences Between Groups or Conditions In this section, we revisit the two basic forms of statistical relationship introduced earlier in the bookdifferences between groups or conditions and relationships between quantitative variablesand we consider how to describe them in more detail. Another reason is that it helps us weight prediction errors and so figure out how to minimize those errors. The Oster data, while free to download, would require special permissions to redistribute. OLS takes that number, squares it into a \(4\), then adds up all the predictions across all your data. When this is not the case, it is said to be skewed. This is the strongest possible negative relationship. In this section, we revisit the two basic forms of statistical relationship introduced earlier in the bookdifferences between groups or conditions and relationships between quantitative variablesand we consider how to describe them in more detail. Its almost like she had other purposes for her study besides providing good examples for my textbook. Real Econometrics. A note is that for estimation type of studies/surveys, sample size calculation needs to consider some other factors too. As we saw earlier in the book, the strength of a correlation between quantitative variables is typically measured using a statistic called Pearsonsr. AsFigure 12.9 shows, its possible values range from 1.00, through zero, to +1.00. Since the line is estimated using all the data, rather than just local data, the results are more precise. the contents by NLM or the National Institutes of Health. Correlation describes an association between variables: when one variable changes, so does the other. In the exposure condition, the children actually confronted the object of their fear under the guidance of a trained therapist. The 25th percentile is the data point which divides the group between the first one-fourth and the last three-fourth of the data. (This was one of several dependent variables.) The correlation coefficient can only range from \(-1\) to \(1\), and the interpretation is the same no matter what units the original variables were in. Nominal/categorical variables are, as the name suggests, variables which can be slotted into different categories (e.g., gender or type of psoriasis). We lose the ability to interpret the slope in terms of the units of \(X\) and \(Y\).6969 Why? In fact, Pearsonsrfor this restricted range of ages is 0. Overview of Non-Experimental Research, 42. ), Hyde points out that although men and women differ by a large amount on some variables (e.g., attitudes toward casual sex), they differ by only a small amount on the vast majority. In this case, we get a value of 45. \tag{4.3} Clinician Rating of Severity: 5.56, Last Name Quartile: First. [3,4] Descriptive statistics give a summary about the sample being studied without drawing any inferences based on probability theory. Draw the LOESS and linear regression curves, # geom_smooth by default draws a LOESS; we don't want standard errors, # 5. A correlational research design investigates relationships between variables without the researcher controlling or manipulating any of them. But besides that. Since the line has \(4X\) in it, we can say that a one-unit increase in \(X\) is associated with a four-unit increase in \(Y\). Otherwise, the larger mean is usuallyM1and the smaller meanM2so that Cohensdturns out to be positive. Plus, its rather choppy. Nonlinear regression is commonly used when \(Y\) can only take a limited number of values. The easiest way to take conditional conditional means is with regression. The natural sciences that emerged as a result of these efforts embody two main components: the scientific knowledge itself, and the ways in which knowledge can be acquired. A correlation coefficient, often expressed as r, indicates a measure of the direction and strength of a relationship between two variables. American Psychological Association (APA) Style, 49. Finally, we need an idea of the expected/crude prevalence either based on previous studies or based on estimates. This means that it is important to make a scatterplot and confirm that a relationship is approximately linear before using Pearsonsr. The other is when one or both of the variables have a limited range in the sample relative to the population. One way of differentiating between continuous and discrete variables is to use the mid-way test. In Figure 4.7, we repeat the vitamin E/BMI relationship from before but now have a straight line fit to it. In the previous chapter, on describing variables, we did a pretty good job covering a lot of what youd want to know when describing a variable. There is a strong negative relationship between age and enjoyment of hip-hop, as evidenced by these ordered pairs: (20, 8), (40, 6), (69, 4), (80, 3). Y = \beta_0 + \beta_1X + \beta_2Z (2005). Oxford University Press.). Or perhaps one of the variables is categorical and theres not really a higher or lower, just different (older children are more likely to use a bike for transportation than younger children). Finally, take the mean of the cross-products. High kurtosis (positive kurtosis also called leptokurtic), Low kurtosis (negative kurtosis also called Platykurtic). As we saw earlier, there are two common situations in which the value of Pearsonsrcan be misleading. \end{equation}\], \[\begin{equation} Let us enter the land of the unexplained. We used SPSS 26.0 for data analysis and exploratory research. We can call the relationship between height and age positive, meaning that for higher values of one of the variables, we expect to see higher values of the other, too (more age is associated with more height). Linear Relationship: A linear relationship will have all the points close together and no curves, dips, etc. Indeed Cohens d values should always be positive so it is the absolute difference between the means that is considered in the numerator. The computations for Pearsonsrare more complicated than those for Cohens d. Although you may never have to do them by hand, it is still instructive to see how. The data is as such: Figure 12.7 long description: Scatterplot showing students scores on the Rosenberg Self-Esteem Scale when scored twice in one week. Identify, define, and describe each of the three main criteria for causality. The research design is divided into three steps: pre-interpretation preparation, consecutive interpretation, and . LOESS provides a local prediction, which it gets by fitting a different shape for each value on the \(X\) axis, with the estimation of that shape weighting very-close observations more than kind-of close observations. What to do? You can read that per as divided by. When we multiply by the standard deviation of \(X\), thats in units of \(X\), so the units cancel out with the per-units-of-X, leaving us with just units-of-Y. If they have nothing to do with each other, then multiplying one by the other will give a positive result about half the time and a negative result the other half, canceling out in step (c) and give you a covariance of \(0\). Since BMI is continuous, Ive cut it up into ten equally-sized ranges (bins) and calculated the proportion taking vitamin E within each of those ranges. In our previous example, since we have already arranged the values in ascending order we find that the point which divides it into two equal halves is the 8th value 42. Then when we divide by the standard deviation of \(Y\), thats in units of \(Y\), canceling out with units-of-\(Y\) and leaving us without any units. 1 The scientific method involves the following steps: Forming a question Performing background research Creating a hypothesis Designing an experiment Collecting data In addition to his guidelines for interpreting Cohensd, Cohen offered guidelines for interpreting Pearsonsrin psychological research (seeTable 12.4). Qualitative research may create theories that can be tested quantitatively. It clearly shows how response time tends to decline as peoples last names get closer to the end of the alphabet. For the first value (30), the deviation from the mean will be 15; for the last value (86), the deviation will be 41. Scatterplots are used when the variable on thex-axis has a large number of values, such as the different possible self-esteem scores. There are two common situations in which the value of Pearsonsrcan be misleading. Want to create or adapt books like this? (The difference in talkativeness discussed inChapter 1 was also trivial:d= 0.06.) Simultaneous administration of the Rosenberg Self-Esteem Scale in 53 nations: Exploring the universal and culture-specific features of global self-esteem. This is the strongest possible positive relationship. Whats the 95th percentile of vitamin E taking overall and for smokers? In other words, were getting the Mean of \(Y\) conditional on \(X\) all conditional on \(Z\). ), Hyde points out that although men and women differ by a large amount on some variables (e.g., attitudes toward casual sex), they differ by only a small amount on the vast majority. Here the points represent individuals, and we can see that the higher students scored on the first occasion, the higher they tended to score on the second occasion. Finally, we get the mean of the shorts-wearing residual conditional on the ice cream residual. In the other hand, descriptive. After all, I cant give you the proportion taking vitamin E among those making $84,325 per year because theres unlikely to be more than one person with that exact number. Because we can also think of the residual as the part of \(Y\) that has nothing to do with \(X\). Such relationships are often presented using line graphs or scatterplots, which show how the level of one variable differs across the range of the other. Do we count you equally if youre very close vs.kind of close? The 75th percentile is the data point which divides the distribution into a first three-fourth and last one-fourth (the last one-fourth being the fourth quartile). While there are no absolute rules, the minimal levels accepted are 0.05 for (corresponding to a significance level of 5%) and 0.20 for (corresponding to a minimum recommended power of 1 0.20, or 80%). Correlation, specifically Pearsons correlation coefficient, takes this exact concept and just rescales it, multiplying the OLS slope by the standard deviation of \(X\) and dividing it by the standard deviation of \(Y\). It is important to calculate the sample size much in advance, rather than have to go for post hoc analysis. Or slope down just a little like in Figure 4.1? It depicts a slightly negative relationship between the variables on the x- and y-axes. The mean fear rating in the control condition was 5.56 with a standard deviation of 1.21. We then split out the distribution by whether someone has engaged in vigorous exercise in the last month. Both of these examples are also linear relationships, in which the points are reasonably well fit by a single straight line. Just off the bat, perhaps we dont just want to know the variation in vitamin E alone. What is a variable? INTRODUCTION As a result of their natural curiosity, human beings seek to understand the environment in which they live and to acquire new knowledge. How does it do this?6666 Calculus, for one. Of course, while this approach is simple and illustrative, its also fairly arbitrary. Continuous variables, on the other hand, can take any value in between the two given values (e.g., duration for which the weals last in the same sample of patients with urticaria). In basic forms of regression, that shape is a straight line. Table 6.1 shows the distribution and the calculations for the data in Example 6.1. The mean of these cross-products, shown at the bottom of that column, is Pearsons, , which in this case is +.53. Figure 4.10 shows the \(X\)-\(Y\) axis on the top-left. So for a one-unit increase in BMI wed expect a .002 increase in the conditional mean of vitamin E. Since vitamin E is a binary variable, we can think of a .002 increase in conditional mean as being a .2 percentage point increase in the proportion of people taking vitamin E. Then, since the standard deviation of taking vitamin E is .369 and the standard deviation of BMI is 6.543, the Pearson correlation between the two is \(.002\times 6.543 / .369 = .355\). Whats the median? Make a scatterplot for these data, compute Pearsons, Condition: Education. Imagine, for example, a study showing that a group of exercisers is happier on average than a group of nonexercisers, with an effect size ofd = 0.35. A control variable is a variable that must be kept constant during the course of an experiment. In this graph, we look at the distribution of how much vitamin E someone takes, among people who take any. So we would get the mean of ice cream conditional on temperature, and then take the residual, getting only the variation in ice cream that has nothing to do with temperature. For one variable that just involves dividing the count in each category by the total to get the proportion - and then converting those to percents by multiplying the proportions by 100% (if percents are desired). The second is 1.58 multiplied by 1.19, which is equal to 1.88. Response Time: 0.2, Last Name Quartile: Second. Describe how relationships between dependent and independent variables influence selection of statistical approach Compare questions of difference, association, and description Identify both basic and complex statistical approaches specific to a research question Basic Approaches to Gathering and Analyzing Quantitative Data If there is a treatment group and a control group, the treatment group mean is usuallyM1and the control group mean isM2. Some variables that might be related to both taking vitamin E and to BMI are gender and age. Lets start with a more basic version - conditional probability. Kurtosis is a representation of outliers. A conditional distribution is the distribution of one variable given the value of another variable. You can do this with install.packages('X') in R, or using a package manager like pip or conda in Python. A pie chart helps show how a total quantity is divided among its constituent variables. Instead of looking at how large the doses are, lets look at whether someone takes vitamin E at all! In research, variables are any characteristics that can take on different values, such as height, age, temperature, or test scores. Run a linear regression, by itself and including controls, # k5 is number of kids under 5 in the house, The Effect: An Introduction to Research Design and Causality, Conditional Conditional Means, a.k.a. using This is roughly saying of all the variation in \(X\), how much of it varies along with \(Y\)?6868 The sheer intuitive nature of this calculation might give a clue as to why we focus on minimizing the sum of squared residuals rather than, say, the residuals to the fourth power, or the product, or the sum of the absolute values. You can plug in a value of \(X\) to get the conditional mean of \(Y\). The scatterplot inFigure 12.7, which is reproduced fromChapter 5, shows the relationship between 25 research methods students scores on the Rosenberg Self-Esteem Scale given on two occasions a week apart. Assume, for example, that there is a strong negative correlation between peoples age and their enjoyment of hip hop music as shown by the scatterplot inFigure 12.10. In case of a total number of values being even, we choose the two middle points and take an average to reach the median. One approach is to use a range of values for the variable were conditioning on rather than a single value. The higher the value of the variable on the x-axis, the lower the value of the variable on the y-axis. Practical Strategies for Psychological Measurement, American Psychological Association (APA) Style, Writing a Research Report in American Psychological Association (APA) Style, From the Replicability Crisis to Open Science Practices. The standard deviation in this formula is usually a kind of average of the two group standard deviations called the pooled-within groups standard deviation. They can be handy for getting a good look at the data and trying to visualize from them what kind of relationship the two variables have. The data presented inFigure 12.7 provide a good example of a positive relationship, in which higher scores on one variable tend to be associated with higher scores on the other (so that the points go from the lower left to the upper right of the graph). Computationally, Pearsonsris the mean cross-product ofzscores. To compute it, one starts by transforming all the scores tozscores. The mid-way value would be 11.5 min which makes sense. It also tells us that the mean of \(Y\) conditional on a given value of \(X\) would be \(4\) higher if you instead made it conditional on a value of \(X\) one unit higher. I could write a whole extra chapter just on cool stuff going on under the hood of OLS. Since there is no clear pattern, the correlation for 18- to 24-year-olds is 0. Nonlinear relationships are not uncommon in psychology, but a detailed discussion of them is beyond the scope of this book. For continuous variables, it is generally better to use groups in the frequency table. For example, researchers Kurt Carlson and Jacqueline Conard conducted a study on the relationship between the alphabetical position of the first letter of peoples last names (from A = 1 to Z = 26) and how quickly those people responded to consumer appeals (Carlson & Conard, 2011)[4]. For example, the frequency distribution in a sample population of males and females can be illustrated as given in Figure 1. Nonlinearrelationshipsare those in which the points are better fit by a curved line. Hoeks S, Kardys I, Lenzen M, van Domburg R, Boersma E. Tools and techniques Statistics: Descriptive statistics. You can also say among all Sarahs, what proportion are women? We would say that this is the probability that someone is a woman conditional on being named Sarah.. Figure out how you want your data to move. d positive linear association with one deviation . A sample size that is too less may make the study underpowered, whereas a sample size which is more than necessary might lead to a wastage of resources. Also, while I repeatedly mention conditional means in this section, there are versions of line-fitting that give conditional medians or percentiles or what-have-you as well. There are two approaches we can take here. One or both of the variables have a limited range in the sample relative to the population. The problem is that some extreme values (outliers), like '86, in this case can skew the value of the mean. For example, in a clinical trial for a topical treatment in psoriasis, the concomitant use of moisturizers might be a confounding variable. Whats the standard deviation of mortality for people who take 90th-percentile levels of vitamin E, and for people who take 10th-percentile levels? Then to the right you can see the \(Z\)-\(Y\) axis, and below the \(Z\)-\(X\) axis. Describing Relationships between Two Variables Up until now, we have dealt, for the most part, with just one variable at a time. What this means in effect is that reducing the effect size will lead to an increase in the required sample size. A Model of Scientific Research in Psychology, 13. However, we gain the ability to more easily tell how strong the relationship is. But theyre close enough in most applications. When one or both variables have a limited range in the sample relative to the population, making the value of the correlation coefficient misleading. What is the relationship between the two variables represented in the table? Another way is graphically. Thus a Cohens dvalue of 0.50 represents a medium-sized difference between two means, and a Cohensdvalue of 1.20 represents a very large difference in the context of psychological research. To calculate the variance, this problem is overcome by adding squares of the deviations. Descriptive research is research designed to provide a snapshot of the current state of affairs. The computations for Pearsonsrare more complicated than those for Cohens d. Although you may never have to do them by hand, it is still instructive to see how. What if we take the explained part out of two different variables? The formula looks like this: Table 12.5 illustrates these computations for a small set of data. The scatterplot shows a diagonal line of points from the bottom left corner to the top right corner. Osters hypothesis is that people who take vitamin E at all should be more likely to do other healthy things like exercise, because both are driven by how health-conscious you are. They are linked with dependent and independent variables and can cause spurious association. The term correlation ratio (eta) is sometimes used to refer to a correlation between variables that have a curvilinear relationship. In a study that compares two groups, a null hypothesis assumes that there is no significant difference between the two groups, and any observed difference being due to sampling or experimental error. We can see that the relationship between the taking of vitamin E and the recommendation being in place is positive (the proportion taking vitamin E is higher during the recommendation time). We can start off with an example of a very straightforward way of showing the relationship between two continuous variables, which is a scatterplot, as shown in Figure 4.1. The relationship between variables determines how the right conclusions are reached. Lets expand our analysis to include a third variable. The range between the 25th percentile and 75th percentile is called the interquartile range. Describing the Relationship between Two Variables Key Definitions Scatter Diagram: A graph made to show the relationship between two different variables (each pair of x's and y's) measured from the same equation. A Cohensdof 0.50 means that the two group means differ by 0.50 standard deviations (half a standard deviation). Creating a research design means making decisions about: Your overall research objectives and approach. In one study, they sent e-mails to a large group of MBA students, offering free basketball tickets from a limited supply. In this case, we consider other values like the median, which is the point that divides the distribution into two equal halves. A quartile is one of the values which break the distribution into four equal parts. This is the key distinction between a simple correlational relationship and a causal relationship. Descriptive statistics can be broadly put under two categories: Sorting and grouping is most commonly done using frequency distribution tables. The horizontal axis is labelled Last Name Quartile, and the vertical axis is labelled Response Times (z Scores) and ranges from 0.4 to 0.4. Asking it to estimate the \(\beta\) values in \(Y = \beta_0 + \beta_1X\) is fine, as before. Pearsonsrhere is .77. Research questions focus on three perspectives: language choice, the use of symbols and their influence on interpreting quality, and the problems in note-taking. Endacott R, Botti M. Clinical research 3: Sample selection. Although post hoc power can be analyzed, a better approach suggested is to calculate 95% confidence intervals for the outcome and interpret the study results based on this. In Stata, packages dont need to be loaded each time theyre used like in R or Python, so Ill always specify in the code example if theres a package that might need to be installed. From Figure 4.6 we can see a clear relationship, with higher values of BMI being associated with more people taking vitamin E. The relationship is very strong at first, but then flattens out a bit, although it remains positive.6161 Why doesnt this dip down at the end like Figure 4.5? If the mean of \(Y\) conditional on \(X = 5\) is \(10\), and we get an observation with \(X = 5\) and \(Y = 13\), then the prediction is \(10\) and the residual is \(13 - 10\). Hyde, J. S. (2007). We determined that the conditional mean of \(Y\) when \(X = 5\) was \(3 + 4(5) = 23\). Hazra A, Gogtay N. Biostatistics series module 5: Determining sample size. This problem is referred to asrestrictionofrange. For example, weve been using all kinds of line-fitting approaches for the relationship between vitamin E and BMI, but vitamin E is binary - you take it or you dont. Three people who get 8 hours of sleep scored 5, 6, and 7 on the depression scale. There are other formulas for computing Pearsonsrby hand that may be quicker. Things get real interesting when we look at the residuals of two variables at once. So, then, how do we do it? Ordinary Least Squares (OLS) is the most well-known application of line-fitting. The relationship between two variables shows you what learning about one variable tells you about the other. So lets do those code examples! Informally, however, the standard deviation of either group can be used instead. Two essential aspects we must understand are the concept of Type I and Type II errors. (This was one of several dependent variables.) Differences Between Groups or Conditions Carlson, K. A., & Conard, J. M. (2011). So how does regression do this? Otherwise, the larger mean is usuallyM1and the smaller meanM2so that Cohensdturns out to be positive. \[\begin{equation} When you get the mean of \(Y\) conditional on \(X\), no matter how you actually do it, youre splitting each observation into two parts - the part explained by \(X\) (the conditional mean), and the part not explained by \(X\) (the residual).

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