Nonlinear Relationships and Graphs without Numbers
A Study on the nonlinear relationship between two-way FDI and industrial structure in China. Jia Nisha1,2 and Zhao Yifan2. Published under licence by IOP . A nonlinear relationship is a type of relationship between two entities What Is the Difference Between a Direct and an Inverse Relationship?. In statistics, dependence or association is any statistical relationship, whether causal or not, The population correlation coefficient ρX,Y between two random variables X The Pearson correlation is +1 in the case of a perfect direct ( increasing) . RDC is invariant with respect to non-linear scalings of random variables.
The polychoric correlation is another correlation applied to ordinal data that aims to estimate the correlation between theorised latent variables.
Correlation and dependence
One way to capture a more complete view of dependence structure is to consider a copula between them. The coefficient of determination generalizes the correlation coefficient for relationships beyond simple linear regression to multiple regression. Sensitivity to the data distribution[ edit ] Further information: This is true of some correlation statistics as well as their population analogues.
Most correlation measures are sensitive to the manner in which X and Y are sampled. Dependencies tend to be stronger if viewed over a wider range of values.
Several techniques have been developed that attempt to correct for range restriction in one or both variables, and are commonly used in meta-analysis; the most common are Thorndike's case II and case III equations.
For example, the Pearson correlation coefficient is defined in terms of momentsand hence will be undefined if the moments are undefined. Measures of dependence based on quantiles are always defined.
- A Study on the nonlinear relationship between two-way FDI and industrial structure in China
- Statistica Software
- Linear, nonlinear, and monotonic relationships
Sample-based statistics intended to estimate population measures of dependence may or may not have desirable statistical properties such as being unbiasedor asymptotically consistentbased on the spatial structure of the population from which the data were sampled. Sensitivity to the data distribution can be used to an advantage. For example, scaled correlation is designed to use the sensitivity to the range in order to pick out correlations between fast components of time series.
If you are asked to construct graphs, and to show a knowledge of graphing by choosing variables yourself, choose carefully what you decide to study. Here is a good example of a difficulty to avoid.
Could you, for example, show a graphical relationship between good looks and high intelligence? I don't think so. First of all, you would have a tough time quantifying good looks though some social science researchers have tried! Intelligence is even harder to quantify, especially given the possible cultural bias to most of our exams and tests.
Finally, I doubt if you could ever find a connection between the two variables; there may not be any. Choose variables that are quantifiable. Height and weight, caloric intake and weight, weight and blood pressure, are excellent personal examples. The supply and demand for oil in Canada, the Canadian interest rate and planned aggregate expenditure, and the Canadian inflation rate during the past forty years are all quantifiable economic variables.
You also need to understand how to plot sets of coordinate points on the plane of the graph in order to show relationships between two variables. One set of coordinates specify a point on the plane of a graph which is the space above the x-axis, and to the right of the y-axis. For example, when we put together the x and y axes with a common origin, we have a series of x,y values for any set of data which can be plotted by a line which connects the coordinate points all the x,y points on the plane.
Such a point can be expressed inside brackets with x first and y second, or 10,1. A set of such paired observation points on a line or curve which slopes from the lower left of the plane to the upper right would be a positive, direct relationship. A set of paired observation or coordinate points on a line that slopes from the upper left of the plane to the lower right is a negative or indirect relationship.
Working from a Table to a Graph Figures 5 and 6 present us with a table, or a list of related numbers, for two variables, the price of a T-shirt, and the quantity purchased per week in a store. Note the series of paired observation points I through N, which specify the quantity demanded x-axis, reflecting the second column of data in relation to the price y-axis, reflecting first column of data. See that by plotting each of the paired observation points I through N, and then connecting them with a line or curve, we have a downward sloping line from upper left of the plane to the lower right, a negative or inverse relationship.
We have now illustrated that as price declines, the number of T-shirts demanded or sought increases. Or, we could say reading from the bottom, as the price of T-shirts increases, the quantity demanded decreases.
We have stated here, and illustrated graphically, the Law of Demand in economics. Now we can turn to the Law of Supply.How To Solve Systems of Nonlinear Equations With Two Variables
The positive relationship of supply is aptly illustrated in the table and graph of Figure 7. Note from the first two columns of the table that as the price of shoes increases, shoe producers are prepared to provide more and more goods to this market. The converse also applies, as the price that consumers are willing to pay for a pair of shoes declines, the less interested are shoe producers in providing shoes to this market.
The x,y points are specified as A through to E. When the five points are transferred to the graph, we have a curve that slopes from the lower left of the plane to the upper right.
We have illustrated that supply involves a positive relationship between price and quantity supplied, and we have elaborated the Law of Supply.
Now, you should have a good grasp of the fundamental graphing operations necessary to understand the basics of microeconomics, and certain topics in macroeconomics. Many other macroeconomics variables can be expressed in graph form such as the price level and real GDP demanded, average wage rates and real GDP, inflation rates and real GDP, and the price of oil and the demand for, or supply of, the product.
Don't worry if at first you don't understand a graph when you look at it in your text; some involve more complicated relationships. You will understand a relationship more fully when you study the tabular data that often accompanies the graph as shown in Figures 5 and 7or the material in which the author elaborates on the variables and relationships being studied.
Gentle Slopes When you have been out running or jogging, have you ever tried, at your starting pace, to run up a steep hill?
If so, you will have a good intuitive grasp of the meaning of a slope of a line. You probably noticed your lungs starting to work much harder to provide you with extra oxygen for the blood.
If you stopped to take your pulse, you would have found that your heart is pumping blood far faster through the body, probably at least twice as fast as your regular, resting rate. The greater the steepness of the slope, the greater the sensitivity and reaction of your body's heart and lungs to the extra work.
Slope has a lot to do with the sensitivity of variables to each other, since slope measures the response of one variable when there is a change in the other. The slope of a line is measured by units of rise on the vertical y-axis over units of run on the horizontal x-axis. A typical slope calculation is needed if you want to measure the reaction of consumers or producers to a change in the price of a product.
For example, let's look at what happens in Figure 7 when we move from points E to D, and then from points B to A.
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The run or horizontal movement is 80, calculated from the difference between and 80, which is Let's look at the change between B and A. The vertical difference is again 20 - 80while the horizontal difference is 80 - We can generalize to say that where the curve is a straight line, the slope will be a constant at all points on the curve. Figure 8 shows that where right-angled triangles are drawn to the curve, the slopes are all constant, and positive.
Now, let's take a look at Figure 9, which shows the curve of a negative relationship. All slopes in a negative relationship have a negative value. We can generalize to say that for negative relationships, increases in one variable are associated with decreases in the other, and slope calculations will, therefore, be of a negative value.
A final word on non-linear slopes.
Linear, nonlinear, and monotonic relationships - Minitab Express
Not all positive nor negative curves are straight lines, and some curves are parabolic, that is, they take the shape of a U or an inverted U, as is demonstrated in Figure 10, shown below. To the left of point C, called the maxima, slopes are positive, and, to the right of point C, they are negative. You can determine the slope of a parabola by drawing a tangent touches at a single point line to any point on the curve.
You can see below that a point such as R is then selected on the line, and a right angled triangle can be constructed which joins points R and B. We can then calculate the rise over the run between points B and R from the distance of the height and the base of the triangle.
So, we can generalize to say that the slopes of a non-linear line are not constant like a straight line and will vary in sign and in value. You will find that a knowledge of slope calculations enhances your understanding of the dynamics of graphs.
It will likely improve your marks in economics, since many test questions require you to illustrate your thinking with graphs. Summary A person from an Eastern culture once observed, "A picture is worth a thousand words.
Without them, we would be forced to examine thousands, or tens of thousands, of bits of statistical information to determine economic relationships. Many economic researchers over the years have done that work for you, and it gets expressed in nice little packages called graphs.