# Curvilinear relationship between two variables hypothesis

### Curvilinear Relationship - SAGE Research Methods

Hypotheses about interaction effects between con- tinuous variables are relationship between two variables is indeed associ- ated with their product. A Curvilinear Relationship is a type of relationship between two variables where as one variable increases, so does the other variable, but only up to a certain. Curvilinear relationship between two variables is positive up to a point and from Normal Distribution, Null hypothesis, Statistical hypothesis testing, Statistical.

Hence, it becomes important to not only study the consequences of CSE, but also its day-to-day antecedents; an endeavor that will significantly increase our understanding of the mechanisms underlying the elicitation and functioning of CSE at work.

On a practical level, conceptualizing CSE as a construct that is subject to within-person variation might open the door for job re design that takes into account these within-individual fluctuations or for various types of managerial interventions aimed at increasing employee CSE. In the present paper, we aim to expand our understanding of the mechanisms underlying day-to-day fluctuations in CSE by examining a how day-to day variation in work pressure is related to day-to day variation in CSE, b how variable, within-person differences in CSE dynamically interact with stable, between-person differences in CSE, and c how within- and between-person differences in CSE together relate to job performance.

This is not surprising as self-efficacy, self-esteem, and neuroticism—all being sub-dimensions of CSE— have been shown to consist of a stable, between- as well as a variable, within-person component Heatherton and Polivy, ; Bandura, ; McNiel and Fleeson, ; Debusscher et al.

Thus, even though individuals are inclined to habitually view themselves in a more positive or negative light, recent research suggests that their self-evaluations vary across time and in different circumstances Judge and Kammeyer-Mueller, ; an idea that closely aligns with the new framing in personality psychology that focuses not only on between- but also on within-person fluctuations Fleeson, ; Funder, In line with this, the present study aims to reconcile the stable trait and the variable state perspectives by examining how state and trait CSE dynamically interact in daily working life.

Turning to the interplay between trait and state CSE, Judge et al. The reason for focusing on work pressure as an antecedent and task performance as an outcome of state CSE is threefold. First, work pressure and task performance are everyday constituents of working life Minbashian et al.

Second, they are elements of all working environments, and therefore they generalize across tasks and situations.

Third, research shows that a stressful working environment relates to correlates of CSE, such as stress, anxiety Wood et al. In what follows, we will first discuss the within-person relationships between work pressure, state CSE, and task performance, and subsequently, we will discuss the moderating effect of trait CSE. Relating Work Pressure to Task Performance: The Mediating Role of State CSE Recent research suggests that not all job demands are alike and that it is important to distinguish between hindrance and challenge demands LePine et al.

Hindrance demands, such as role conflict or red tape, are typically perceived as opposing personal growth and achievement, which implies that, even if employees are able to overcome them, they offer little to no potential gain Cavanaugh et al. Instead, challenge demands such as task complexity and work pressure are perceived by employees as opportunities to learn and achieve, and therefore they create an opportunity for personal growth and goal achievement Cavanaugh et al.

However, besides their motivational effect, challenge demands are also energy-draining, manifested in the positive relationship with psychological strain and ill health Boswell et al. The theory and empirical research on challenge demands suggest that work pressure has the potential to stimulate as well as deplete work outcomes; a dual function that is supported by an inverted U-shaped relationship between challenge demands on one hand and performance, motivation, job satisfaction, and other important work outcomes on the other hand Xie and Johns, ; De Jonge and Schaufeli, ; Zivnuska et al.

To explain this curvilinear relationship, researchers often draw on the Yerkes-Dodson law Yerkes and Dodson, and activation theory Gardner, ; Gardner and Cummings, Both theories suggest that at very low levels of activation, people are apathetic. Therefore, increases in work-related stimulation have an energizing effect when the current stimulation level is low.

Therefore, drawing on the Yerkes-Dodson law Yerkes and Dodson, and activation theory Gardner, ; Gardner and Cummings,we expect within-person variation in work pressure to relate to within-person variation in task performance in an inverted U-shaped way. Work pressure has an inverted U-shaped within-person relationship with task performance.

In particular, when working under little work pressure, people may feel in control, but at the same time they might feel under-stimulated, frustrated, and passive Gardner, ; Gardner and Cummings, ; Zivnuska et al. As a result of this mixture of experiences, their state CSE will be sub-optimal.

This mixture of ingredients i. Finally, when work pressure grows further it might become overwhelming, there by depleting the sense of self-efficacy and self-worth, evoking the feeling that the person is no longer in control, and boosting state neuroticism because of increased feelings of anxiety Zivnuska et al. This idea of a curvilinear relationship between job demands and how one acts, feels, and thinks has been supported by research showing that challenge stressors relate curvilinearly to anxiety and emotional exhaustion Xie and Johns, ; De Jonge and Schaufeli, In summary, we suggest that the relationship between work pressure and state CSE is inverted U-shaped; it peaks at moderate levels and declines at low and high levels of work pressure.

Work pressure has an inverted U-shaped within-person relationship with state CSE. Although there is to the best of our knowledge only one within-person study on the positive relationship between CSE and task performance Debusscher et al. An important reason for the positive relationship between CSE and task performance is that individuals who are high on CSE are better at setting goals, working toward them, and are as a result more motivated to perform their jobs.

Indeed, both in a lab experiment and a field study, Erez and Judge demonstrated that CSE related to task motivation, persistence, goal setting, goals commitment, activity level, and task performance. Building on these findings, we hypothesize that day-to day variation in state CSE relates positively to day-to day variation in task performance, which, when combined with the foregoing hypotheses, implies that state CSE is expected to mediate the curvilinear within-person relationship between work pressure and task performance.

State CSE mediates the inverted U-shaped within-person relationship between work pressure and task performance. This expectation follows from the conceptualization of traits as individual differences in the sensitivity to situational provocation. Building on the idea of traits as situational sensitivities, we argue that trait CSE relates to contingent units of CSE i. That is, for a person high in trait CSE, we expect the level of state CSE to be less contingent upon the level of work pressure because they are less susceptible to it.

This reasoning is in line with the finding that people high in trait neuroticism react more strongly to negative environmental features than people low in neuroticism, even when confronted with relatively small problems Suls and Martin, ; Debusscher et al. In the same vein, Bolger and Schilling demonstrated that people high in trait neuroticism have an increased reactivity to stressful situations.

Finally, for self-esteem, it has been shown that people high in trait self-esteem are protected from the effects of external factors Mossholder et al.

As emotional stability being the counterpart of neuroticismhigh self-esteem, and high self-efficacy are indicators of high CSE, these findings suggest that people high in trait CSE might be less susceptible to variation in work pressure than low trait CSE people.

Materials and Methods Participants Fifty-five employees 33 women from different Belgian companies participated in the study. On average, respondents were Fifteen participants had a secondary school degree, 12 completed a higher professional education, and 28 completed higher academic education.

In terms of job content, 16 worked in logistics and distribution, 13 in governmental and non-profit organizations, 6 in health care, 6 in telecom, 4 in the financial sector, 1 in chemistry and pharmacy, 3 in human resources, 2 in communication, and 4 in other jobs.

Ten participants worked part-time seven participants worked 4 days, one participant worked 3 days, and two worked 2. We recruited participants in several ways. We posted a call on the intranet of the Flemish education networks, in the alumni newsletter of the Vrije Universiteit Brussel, and we emailed personal contacts.

In these calls, we explained the goal of the study and stressed that the anonymity of records would be ensured. We only contacted people again who indicated that they were willing to participate in the study via email or orally. Participants were enrolled in a day daily diary study in which trait CSE was measured at baseline, while work pressure, state CSE, and task performance were assessed daily.

For the daily diary part, participants received an email each working day including a link to a survey in which they had to report on their level of work pressure, state CSE, and level of task performance, and they did so for 10 consecutive working days.

At the beginning of each survey, we again stressed that the data would be made anonymous.

- Curvilinear Relationship

Moreover, participants could stop participating in the study whenever they wanted. All scales, as well as the items within each scale, were randomized. To allow for a momentary or state measure of CSE, we slightly adapted the items e. Work Pressure Work pressure was measured using the three-item scale of Bakker et al.

Similar to the state CSE scale, we slightly adapted it to allow for daily ratings of work pressure e. Task Performance Task performance was measured using the seven-item task performance subscale of Williams and Anderson Similar to the state CSE scale, we slightly adapted it to allow for momentary self-ratings of performance e.

**TEAS Math Tutorial - The Relationship Between Two Variables**

Analyses Because of the complexity of the mediation model, we first tested all hypothesized relationships separately using two-level regression analyses with the lme4 package in R Bates, All level-1 predictors i. This procedure ensures that the level-1 predictors contain within-person variability only, which is necessary because the hypotheses regarding the relationships between work pressure, state CSE, and task performance pertain to the within-person level.

To test whether the effect of the level-1 predictors was consistent across individuals, we tested whether a model with a random slope on the between-person level fitted our data significantly better than a model without random slopes. Next, the hypotheses were tested simultaneously using Bayesian two-level path modeling in Mplus version 7. Moreover, it allows testing complicated models. An important difference between Bayesian and the more traditional—frequentist—approach is that Bayesian analysis does not yield p-values and confidence intervals.

Instead, for each parameter in the model, Bayesian analysis yields a posterior distribution, which shows the probability distribution of the parameter given the data Kruschke et al. Based on these posterior distributions, credibility intervals can be constructed.

These credibility intervals include a predefined percentage of the posterior distribution e. For our Bayesian analysis, we will draw on these credibility intervals to help deciding which parameter values should be deemed credible or not Kruschke et al. These ICCs show, for each level-1 variable, the proportion of variation due to between- and within-person differences.

Overall, the ICCs show that a substantial part of the variability in work pressure, state CSE, and task performance is due to within-person differences. Means, standard deviations, intra-class correlations and correlations for all study variables. Next, we tested the hypothesized relationships by means of a series of two-level regression analyses.

## Handbook of Biological Statistics

First, we tested whether within-person fluctuations in work pressure relate in an inverted U-shaped way to within-person fluctuations in task performance i. To do so, we predicted momentary task performance from work pressure and work pressure squared work pressure was person-centered before computing the squared effect. Moreover, we tested whether these relationships varied across individuals.

Next, we tested whether there is an inverted U-shaped within-person relationship between work pressure and state CSE i. This was done by adding the main effect of trait CSE as well as the interaction between trait CSE and the linear component of work pressure to the previous model.

A graphical representation of this moderation effect is shown in Figure 2which shows that the level of state CSE of people high on trait CSE is less affected by the level of work pressure these people experience. This could be acceptable if the line is just slightly curved; if your biological question is "Does more X cause more Y?

However, it will look strange if you use linear regression and correlation on a relationship that is strongly curved, and some curved relationships, such as a U-shape, can give a non-significant P value even when the fit to a U-shaped curve is quite good. And if you want to use the regression equation for prediction or you're interested in the strength of the relationship r2you should definitely not use linear regression and correlation when the relationship is curved.

A second option is to do a data transformation of one or both of the measurement variables, then do a linear regression and correlation of the transformed data. There are an infinite number of possible transformations, but the common ones log, square root, square will make a lot of curved relationships fit a straight line pretty well. This is a simple and straightforward solution, and if people in your field commonly use a particular transformation for your kind of data, you should probably go ahead and use it.

If you're using the regression equation for prediction, be aware that fitting a straight line to transformed data will give different results than fitting a curved line to the untransformed data. Your third option is curvilinear regression: There are a lot of equations that will produce curved lines, including exponential involving bX, where b is a constantpower involving Xblogarithmic involving log Xand trigonometric involving sine, cosine, or other trigonometric functions.

For any particular form of equation involving such terms, you can find the equation for the curved line that best fits the data points, and compare the fit of the more complicated equation to that of a simpler equation such as the equation for a straight line. Here I will use polynomial regression as one example of curvilinear regression, then briefly mention a few other equations that are commonly used in biology. A polynomial equation is any equation that has X raised to integer powers such as X2 and X3.

It produces a parabola. You can fit higher-order polynomial equations, but it is very unlikely that you would want to use anything more than the cubic in biology. Null hypotheses One null hypothesis you can test when doing curvilinear regression is that there is no relationship between the X and Y variables; in other words, that knowing the value of X would not help you predict the value of Y.

This is analogous to testing the null hypothesis that the slope is 0 in a linear regression. You measure the fit of an equation to the data with R2, analogous to the r2 of linear regression. A cubic equation will always have a higher R2 than quadratic, and so on.

The second null hypothesis of curvilinear regression is that the increase in R2 is only as large as you would expect by chance.

Assumptions If you are testing the null hypothesis that there is no association between the two measurement variables, curvilinear regression assumes that the Y variable is normally distributed and homoscedastic for each value of X. Since linear regression is robust to these assumptions violating them doesn't increase your chance of a false positive very muchI'm guessing that curvilinear regression may not be sensitive to violations of normality or homoscedasticity either.

I'm not aware of any simulation studies on this, however. Curvilinear regression also assumes that the data points are independentjust as linear regression does. You shouldn't test the null hypothesis of no association for non-independent data, such as many time series.

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However, there are many experiments where you already know there's an association between the X and Y variables, and your goal is not hypothesis testing, but estimating the equation that fits the line. For example, a common practice in microbiology is to grow bacteria in a medium with abundant resources, measure the abundance of the bacteria at different times, and fit an exponential equation to the growth curve. The amount of bacteria after 30 minutes is not independent of the amount of bacteria after 20 minutes; if there are more at 20 minutes, there are bound to be more at 30 minutes.

However, the goal of such an experiment would not be to see whether bacteria increase in abundance over time duh, of course they do ; the goal would be to estimate how fast they grow, by fitting an exponential equation to the data.

For this purpose, it doesn't matter that the data points are not independent. Just as linear regression assumes that the relationship you are fitting a straight line to is linear, curvilinear regression assumes that you are fitting the appropriate kind of curve to your data. If you are fitting a quadratic equation, the assumption is that your data are quadratic; if you are fitting an exponential curve, the assumption is that your data are exponential. Violating this assumption—fitting a quadratic equation to an exponential curve, for example—can give you an equation that doesn't fit your data very well.

In some cases, you can pick the kind of equation to use based on a theoretical understanding of the biology of your experiment. If you are growing bacteria for a short period of time with abundant resources, you expect their growth to follow an exponential curve; if they grow for long enough that resources start to limit their growth, you expect the growth to fit a logistic curve.

Other times, there may not be a clear theoretical reason for a particular equation, but other people in your field have found one that fits your kind of data well. And in other cases, you just need to try a variety of equations until you find one that works well for your data. How the test works In polynomial regression, you add different powers of the X variable X, X2, X3… to an equation to see whether they increase the R2 significantly. The R2 will always increase when you add a higher-order term, but the question is whether the increase in R2 is significantly greater than expected due to chance.

You can keep doing this until adding another term does not increase R2 significantly, although in most cases it is hard to imagine a biological meaning for exponents greater than 3. Even though the usual procedure is to test the linear regression first, then the quadratic, then the cubic, you don't need to stop if one of these is not significant. For example, if the graph looks U-shaped, the linear regression may not be significant, but the quadratic could be.

Examples Fernandez-Juricic et al. They counted breeding sparrows per hectare in 18 parks in Madrid, Spain, and also counted the number of people per minute walking through each park both measurement variables.

Graph of sparrow abundance vs.

### Curvilinear Relationship definition | Psychology Glossary | guiadeayuntamientos.info

This seems biologically plausible; the data suggest that there is some intermediate level of human traffic that is best for house sparrows. Perhaps areas with too many humans scare the sparrows away, while areas with too few humans favor other birds that outcompete the sparrows for nest sites or something. Even though the cubic equation fits significantly better than the quadratic, it's more difficult to imagine a plausible biological explanation for this.