# What is a curvilinear relationship

### Curvilinear Relationship - SAGE Research Methods

We hypothesized that (a) work pressure relates to task performance in a curvilinear way, (b) state CSE mediates the curvilinear relationship between work . Items 1 - 33 of 33 A curvilinear relationship is a type of relationship between two variables that has a pattern of correspondence or association between the two. Define curvilinear correlation. curvilinear correlation synonyms, curvilinear correlation pronunciation, curvilinear correlation translation, English dictionary.

Finally, we tested a model in which momentary task performance was predicted by state CSE, work pressure, and work pressure squared 1. Momentary task performance as a function of work pressure.

The work pressure scores are person-centered. State Core Self-Evaluations as a function of work pressure. Next, we tested the moderated mediation model in its entirety using Bayesian two-level path analysis. To this end, a model was tested in which state CSE was predicted by the linear and squared effect of work pressure, while momentary task performance was predicted from state CSE and the linear and squared effect of work pressure all these relationships were modeled at the within-person level.

Moreover, and in line with the results of the multilevel regression analyses, we included random slopes for the relationship between work pressure and state CSE, the relationship between work pressure and momentary task performance, and the relationship between work pressure squared and momentary task performance. To formally test the indirect mediation effect of work pressure on momentary task performance via state CSE i.

Because the relationship between work pressure X and state CSE i. Because of this reason, Hayes and Preacher refer to the indirect effect as the instantaneous indirect effect, which is the effect of the predictor on the outcome through the mediator s at a specific value of the predictor.

From this figure, it can be seen that for low levels of work pressure the instantaneous indirect effect of work pressure on task performance via state CSE is positive [e. This implies that, when work pressure is low, further increases in work pressure promote task performance via their effect on state CSE. Moreover, because the curves—describing the instantaneous indirect effect— decrease, the motivational effect of increases in work pressure weakens with increased levels of initial work pressure.

On the contrary, for high initial levels of work pressure, the instantaneous indirect effect of work pressure on task performance via state CSE is negative [e.

This means that further increases in work pressure deplete task performance via their negative effect on state CSE.

Moreover, this depleting effect becomes stronger when the initial level of work pressure is higher which can be seen from the fact that the curves decrease.

Combined, Figure 3 thus provides support for a curvilinear mediation effect i. Regarding the moderation effect of trait CSE, Figure 3 shows that for people low in trait CSE the depleting effect of work pressure via state CSE especially holds for low levels of work pressure, while for people high in trait CSE the depleting effect is especially located at high levels of work pressure.

The instantaneous indirect mediation effect of work pressure on momentary task performance via state core self-evaluations CSE as a function of work pressure person-centered values. The left panel shows the mediation effect for people scoring 1 SD below the average on trait CSE; the middle panel shows the mediation effect for people with an average level of trait CSE; and the right panel shows the mediation effect for people scoring 1 SD above the average on trait CSE.

Discussion With the present paper, we contributed to a better understanding of the role of CSE at the workplace. This was done by a shedding light on a work-related trigger i. This is a major contribution to the literature on CSE, as it uncovers the mechanisms through which CSE relates to work outcomes in everyday working life.

In what follows, we will discuss the theoretical and practical implications of our findings. Importantly, our findings not only support, but go well beyond the mechanisms proposed by person-situation interactionist models such as Trait Activation Theory Tett and Guterman, and the Traits as Situational Sensitivities Model Marshall and Brown, by showing that the mediation effect of work pressure on task performance via state CSE is not only quantitatively, but also qualitatively different for people with different levels of trait CSE.

That is, for people low in trait CSE, the depleting effect of work pressure via state CSE operates for low but not for high levels of work pressure, while for people high in trait CSE the depleting effect is located at high but not at low levels of work pressure. We suggest that this dual mechanism can be explained by goal setting Locke and Latham, and self-discrepancy theory Higgins, In particular, low levels of work pressure might not pose a problem for people high in trait CSE because these individuals have a higher level of goal setting motivation Erez and Judge, An important reason for this might be that goal commitment—which is an element of goal setting motivation— is a function of expected goal attainment, and this is per definition higher in people who are high in trait CSE.

Because people high in trait CSE have higher levels of goal commitment, they do not require external pressure to perform well. People low in trait CSE, in contrast, do not have this strong base of resources, and therefore rely more on external pressures to regulate their behavior.

Indeed, because they are less likely to believe that they can achieve what they want to achieve, their level of goal commitment is generally lower. Therefore, their level of state CSE is more strongly influenced by external pressures when the level of work pressure is low. The result of all of this is that under conditions of low work pressure, the level of state CSE of high trait CSE people is virtually unaffected when work pressure decreases, while the level of state CSE of low trait CSE people decreases because of the combination of under-stimulation and a lack of goal setting motivation.

Turning to high levels of work pressure, we believe that the reason for the detrimental effect of increased levels of work pressure on the state CSE of individuals high on trait CSE, may be that their self-image strongly relies on the idea that they succeed in whatever they undertake.

However, when they come across a situation in which the level of work pressure is too high, this high sense of achievement gets threatened, which, according to self-discrepancy theory Higgins,leads to a flow of negative emotions such as disappointment, dissatisfaction, sadness, and depression.

People with a low trait CSE level, in turn, should experience these feelings of self-discrepancy to a lesser extent because for them not being able to cope with the demands at hand is nothing new, and is more congruent with their self-image. As a result, under high work pressure, the level of state CSE of people high on trait CSE decreases when high work pressure increases further due to increasing feelings of self-discrepancy, while the state CSE level of people low on trait CSE does not decrease substantially because being unable to meet demands is not perceived as a shock for their self-image.

It should be noted that, to formally test this dual mechanisms account, future research is needed in which goal commitment and self-discrepancy are measured along with work pressure, state CSE, trait CSE and task performance. A possible alternative explanation for the finding that there are qualitatively different mediation effects for people high and low on trait CSE is that the levels of perceived work pressure might not be comparable.

Because we person-centered the perceived work pressure scores, all between-person differences in work pressure were removed from the data. Yet, it might be that that the baseline of work pressure is higher for trait CSE people, as they seek and create jobs that offer challenges; an idea that aligns with the finding that people select situations that are congruent with their personality Emmons et al.

As a result, for people high on trait CSE, levels of work pressure that are lower than usual can still be relatively high, and therefore they might still be challenging and not be associated with apathy.

Conversely, levels of work pressure higher than usual might be extremely high for people high in trait CSE, which would then lead to overload and depletion of their state CSE level.

For people low in trait CSE, levels of work pressure lower than usual may be very low and therefore offer no stimulation at all, hence depleting their state CSE level.

When experiencing more work pressure than usual, the level of work pressure might be high, but still manageable for those low in trait CSE; and therefore it should not relate to decreased levels of state CSE.

We tested this alternative explanation by regressing state CSE on the grand-mean centered work pressure scores which contain both between- and within-person variability. Although the effects are weaker i.

### Curvilinear Regression

This implies that between-person differences in work pressure cannot fully explain the qualitatively different mechanisms. However, to find a definite answer to the question whether individual differences in the average level of work pressure might explain why people with different trait CSE levels react differently to work pressure, future research is needed.

One way to do so would be to manipulate work pressure rather than to measure it. Practical Implications In line with previous findings on challenge demands, our study shows that, up to some point, work pressure might stimulate state CSE and task performance. This implies that managers should not always try to decrease the level of work pressure. Instead, they might try to keep work pressure at a moderate level as this seems to work best with all employees. Additionally, our findings also revealed that the mechanism relating work pressure to task performance is different for people with different trait CSE levels.

The weights cannot really be interpreted separately. Note also that if we subtract the mean of X from X, then the b weights will change. The increment in variance will not, nor will the graph of the curve. I mention this to underscore the point that you do not interpret the b weights for the variables when you include power terms.

If you want to know the importance of a variable in the predicted or variance accounted for sense, then you need to compute the change in R2 between the model with the linear variable and all power terms absent to the model with the linear variable and all power terms present. They work together as a block, and need to be treated as such. Under no circumstances should you enter linear and power terms in a variable selection routine such as stepwise predictor selection.

Such a practice can lead to nonsense such as concluding that a squared term contributes variance, but the linear term does not. Again, the linear term and associated power terms must be treated as a block. If you want to know about the importance of the variable in an explanatory sense, it is very difficult to figure.

It is hard to include nonlinear terms in path and structural equation models and to interpret them. There is a literature on this, however, that you may read if you need to. Probably the interpretation of the importance of nonlinear relations is best tackled in the context of the particular problem in which you are working.

### Curvilinear regression - Handbook of Biological Statistics

Computing and Interpreting Interactions With two continuous variables, we can have an interaction. An interaction means that the level of one variable influences the effect "importance" of the other variable.

For example, it might be the case that creativity and intelligence interact to produce novel, useful mechanical devices we might have people produce something and have it judged by a panel of experts.

Suppose that the correlation between creativity and productivity gets larger as intelligence increases. For people with little intelligence, high creativity does not lead to useful devices novel, perhaps, but useless as the transporter on the set of Star Trek for actually moving people.

For people with high intelligence there is a strong correlation between creativity and productivity. Note that the regression line for predicting productivity from creativity becomes steeper and the error of prediction is reduced as cognitive ability increases r increases. Such an interaction would be symmetric. For people with little creativity, there would be little or no correlation between intelligence and productivity.

For people with high creativity, there would be a strong correlation between intelligence and productivity. We could create three new graphs to show these relations. All we would have to do is take the graphs we already make and to substitute the terms "creativity" and "cognitive ability. In regression terms, an interaction means that the level of one variable influences the slope of the other variable.

We model interaction terms by computing a product vector that is, we multiply the two IVs together to get a third variableand then including this variable along with the other two in the regression equation.

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.

## Curvilinear Relationship

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.

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.

**Curvilinear regression**

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.