X,Y scatterplotThese are plots of X,Y coordinates showing each individual's or sample's score on two variables.
When plotting data this way we are usually interested in knowing whether the two variables show a "relationship", i.e. do they change in value together in a consistent way?
When comparing one measured variable against another—looking for trends or associations— it is appropriate to plot the individual data points on an x-y plot, creating a scatterplot.
A scatter plot is a type of graph that shows how two sets of data might be connected. When you plot a series of points on a graph, you’ll have a visual idea of whether your data might have a linear, exponential or some other kind of connection.
Creating scatter plots by hand can be cumbersome, especially if you have a large number of plot points. Microsoft Excel has a built in graphing utility that can instantly create a scatter plot from your data. This enables you to look at your data and perform further tests without having to re-enter your data. For example, if your scatter plot looks like it might be a linear relationship, you can perform linear regression in one or two clicks of your mouse.
If the relationship is thought to be linear, a linear regression line can be calculated and plotted to help filter out the pattern that is not always apparent in a sea of dots (Figure 3).
In this example, the value of r (square root of R2) can be used to help determine if there is a statistical correlation between the x and y variables to infer the possibility of causal mechanisms.
Such correlations point to further questions where variables are manipulated to test hypotheses about how the variables are correlated.
Students can also use scatterplots to plot a manipulated independent x-variable against the dependent y-variable. Students should become familiar with the shapesthey’ll find in such scatterplots and the biological implications of these shapes.
A concave upward curve is associated with exponentially increasing functions (for example, in the early stages of bacterial growth).
In ecology, a species-area curve is a relationship between the area of a habitat, or of part of a habitat, and the number of species found within that area.
A sine wave–like curve is associated with a biological rhythm.
A sine wave–like curve is associated with a biological rhythm.
Figure 1: Predator-Prey Curve
Students will usually use computer software to create their graphs. In so doing, theyshould keep in mind the following elements of effective graphing:
• A graph must have a title that informs the reader about the experiment and tells thereader exactly what is being measured.
• The reader should be able to easily identify each line or bar on the graph.
Elements of effective graphing
Big or little? For course-related papers, a good rule of thumb is to size your figures to fill about one-half of a page. Readers should not have to reach for a magnifying glass to make out the details. Compound figures may require a full page
• Axes must be clearly labeled with units as follows:
––The x-axis shows the independent variable. Time is an example of an independent variable. Other possibilities for an independent variable might be light intensity or the concentration of a hormone or nutrient.
––The y-axis denotes the dependent variable— the variable that is being affected by the condition (independent variable) shown on the x-axis.
Intervals must be uniform. For example, if one square on the x-axis equals five minutes, each interval must be the same and not change to 10 minutes or one minute. The intervals do not have to be the same on each axis… they represent different quantities.
If there is a break in the graph, such as a time course over which little happens for an extended period, it should be noted with a break in the axis anda corresponding break in the data line.
Tick marks - Use common sense when deciding on major (numbered) versus minor ticks. Major ticks should be used to reasonably break up the range of values plotted into integer values. Within the major intervals, it is usually necessary to add minor interval ticks that further subdivide the scale into logical units (i.e., a interval that is a factor of the major tick interval). For example, when using major tick intervals of 10, minor tick intervals of 1,2, or 5 might be used, but not 4.
–– It is not necessary to label each interval. Labels can identify every five or 10 intervals, or whatever is appropriate.
––The labels on the x-axis and y-axis should allow the reader to easily see the information.
Parts of a Graph: This is an example of a typical line graph with the various component parts labeled in red.
More than one condition of an experiment may be shown on a graph by the use of different lines. For example, the appearance of a product in an enzyme reaction at different temperatures can be compared on the same graph. In this case, each line must be clearly differentiated from the others—by a label, a different style, or colors indicated by a key. These techniques provide an easy way to compare the results of experiments.
Figure 3: Release of reducing sugars from alfalfa straw by crude extracellular enzymes from thermophilic and nonthermophilic fungi.
• The graph should clarify whether the data start at the origin (0,0) or not. The line should not be extended to the origin if the data do not start there. In addition, the line should not be extended beyond the last data point (extrapolation) unless a dashed line(or some other demarcation) clearly indicates that this is a prediction about what may happen.
A scatterplot is a useful summary of a set of bivariate data (two variables), usually drawn before working out a linear correlation coefficient or fitting a regression line.
It gives a good visual picture of the relationship between the two variables, and aids the interpretation of the correlation coefficient or regression model.
Scatterplot
Each unit contributes one point to the scatterplot, on which points are plotted but not joined. The resulting pattern indicates the type and strength of the relationship between the two variables.
The following plots demonstrate the appearance of positively associated, negatively associated, and non-associated variables:
Positive correlation
Negative correlation
No correlation
A scatterplot can be a helpful tool in determining the strength of the relationship between two variables. If there appears to be no association between the proposed explanatory and dependent variables (i.e., the scatterplot does not indicate any increasing or decreasing trends), then fitting a linear regression model to the data probably will not provide a useful model.
Positive correlation
Negative correlation
No correlation
Correlation Statistics – allow one to determine/describe the relationship between variables.
a. Linear Regression – Line of best fit used to express the relationship between two variables and predict potential outcomes based on a given value for a variable.
i. The line of best fit follows the familiar equation of y = mx + b, where b is the y intercept and m is the
slope of the line.ii. A steep slope indicates a strong effect.iii. A shallow slope indicates a weak effect.iv. A negative slope indicates a negative effect. That is an
increase in X results in a decrease in Y.v. The line of best fit can be used to predict a value of one
variable given a value for the other variable.
Linear regression attempts to model the relationship between two variables by fitting a linear equation to observed data. One variable is considered to be an independent (explanatory) variable, and the other is considered to be a dependent variable. For example, a modeler might want to relate the weights of individuals to their heights using a linear regression model.
Linear Regression
R2 is a statistic that will give some information about the goodness of fit of a model. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1 indicates that the regression line perfectly fits the data.
A valuable numerical measure of association between two variables is the correlation coefficient, which is a value between -1 and 1 indicating the strength of the association of the observed data for the two variables.
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Correlation
positive
A positive correlation indicates a positive association between the variables (increasing values in one variable correspond to increasing values in the other variable), while a negative correlation indicates a negative association between the variables (increasing values is one variable correspond to decreasing values in the other variable). A correlation value close to 0 indicates no association between the variables.
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Correlation in Linear RegressionThe square of the correlation coefficient, R², is a useful value in linear regression. This value represents the fraction of the variation in one variable that may be explained by the other variable. Thus, if a correlation of 0.8 is observed between two variables (say, height and weight, for example), then a linear regression model attempting to explain either variable in terms of the other variable will account for 64% of the variability in the data.
The correlation coefficient also relates directly to the regression line Y = a + bX for any two variables.Because the least-squares regression line will always pass through the means of x and y, the regression line may be entirely described by the means, standard deviations, and correlation of the two variables under investigation.
A linear regression line has an equation of the form:
x is the independent variabley is the dependent variable m is slope of the line is b b is the intercept (the value of y when x = 0)
y =mx+b
The most common method for fitting a regression line is the method of least-squares. This method calculates the best-fitting line for the observed data by minimizing the sum of the squares of the vertical deviations from each data point to the line (if a point lies on the fitted line exactly, then its vertical deviation is 0). Because the deviations are first squared, then summed, there are no cancellations between positive and negative values.
Least-Squares Regression
Given a scatter plot, we can draw the line that best fits the data
There are two tests for correlation:
the Pearson correlation coefficient ( r ), and Spearman's rank-order correlation coefficient (rs ).
These both vary from +1 (perfect correlation) through 0 (no correlation) to –1 (perfect negative correlation).
If your data are continuous and normally-distributed use Pearson, otherwise use Spearman.
What is the Pearson Correlation Coefficient?
Correlation between variables is a measure of how well the variables are related. The most common measure of correlation in statistics is the Pearson Correlation (technically called the Pearson Product Moment Correlation or PPMC), which shows the linear relationship between two variables. Two letters are used to represent the Pearson correlation: Greek letter rho (ρ) for a population and the letter “r” for a sample.
R2 is a statistic that will give some information about the goodness of fit of a model. In regression, the R2 coefficient of determination is a statistical measure of how well the regression line approximates the real data points. An R2 of 1 indicates that the regression line perfectly fits the data.
Correlation between variables is a measure of how well the variables are related. The most common measure of correlation in statistics is the Pearson Correlation (technically called the Pearson Product Moment Correlation or PPMC), which shows the linear relationship between two variables.
Two letters are used to represent the Pearson correlation: Greek letter rho (ρ) for a population and the letter “r” for a sample.
r =n(Σxy)−(Σx)(Σy)
[nΣx2 −(Σx)2 ] [nΣy2 −(Σy)2 ]
In linear least squares regression with an estimated intercept term, R2 equals the square of the Pearson correlation coefficient between the observed and modeled (predicted) data values of the dependent variable.
Results are between -1 and 1. A result of -1 means that there is a perfect negative correlation between the two values at all, while a result of 1 means that there is a perfect positive correlation between the two variables. A result of 0 means that there is no linear relationship between the two variables.
What are the Possible Values for the Pearson Correlation?
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What are the Possible Values for the Pearson Correlation?
You will very rarely get a correlation of 0, -1 or 1. You’ll get somewhere in between. The closer the value of r gets to zero, the greater the variation the data points are around the line of best fit.High correlation: 0.5 to 1.0 or -0.5 to 1.0Medium correlation: 0.3 to 0.5 or -0.3 to 0.5Low correlation: 0.1 to 0.3 or -0.1 to -0.3
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Pearson Product Moment (PPM) Correlation – unit-lessvalue ranging from –1.0 to +1.0 that describes the goodness of fit of the relationship between two variables.
i. An |r| value of 1.00 represents a perfect correlation.
ii. An |r| value above 0.85 represents a very high correlation.
iii. An |r| value of 0.70 – 0.84 represents a high correlation.
iv. An |r| value of 0.55 – 0.69 represents a moderate correlation.
v. An |r| value of 0.40 – 0.54 represents a low correlation.
vi. An |r| value of 0.00 – 0.39 represents no correlation.
In statistics, the Pearson product-moment correlation coefficient (sometimes referred to as the PPMCC or PCC, or Pearson's r) is a measure of the linear correlation (dependence) between two variables X and Y, giving a value between +1 and −1 inclusive. It is widely used in the sciences as a measure of the strength of linear dependence between two variables. It was developed by Karl Pearson from a related idea introduced by Francis Galton in the 1880s.
What Do I Have to Consider When Using the Pearson product-moment correlation?
The PPMC does not differentiate between dependent and independent variables. For example, if you are investigating the correlation between a high caloric diet and diabetes, you might find a high correlation of 0.8. However, you could also run a PPMC with the variables switched around (diabetes causes a high caloric diet), which would make no sense. Therefore, as a researcher you have to be mindful of the variables you are plugging in. In addition, the PPMC will not give you any information about the slope of the line — it only tells you whether there is a high correlation.
Real Life ExamplePearson correlation is used in thousands of real life situations. For example, scientists in China wanted to know if there was a correlation between spatial distribution and genetic differentiation in weedy rice populations in a study to determine the evolutionary potential of weedy rice.
Real Life ExampleThe graph below shows the observed heterozygosity of weedy rice plotted against the multilocus outcrossing rate. Pearson’s correlation between the two groups was analyzed, showing a significant positive correlation of between 0.783 and 0.895 for weedy rice populations.
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Figure 2: Pearson correlation comparing overall rating versus staff rating (n = 4999, Pearson correlation, r = .715, P < .001).
Analysis of 4999 Online Physician Ratings Indicates That Most Patients Give Physicians a Favorable RatingKadry B, Chu LF, Kadry B, Gammas D, Macario A - J. Med. Internet Res. (2011)
f1-ndt-6-009: A) Positive correlation between the Bmax and the cognitive complexity factor in men (Pearson correlation = 0.378, P = 0.006).
B) Negative correlation between the Kd and the motor impulsivity factor in men (Pearson correlation = −0.673, P = 0.023).
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Impulsivity, gender, and the platelet serotonin transporter in healthy subjects
Comparison Between Dynamic Contour Tonometry and Goldmann Applanation Tonometry
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Figure 1: Pearson correlation analysis of intraocular pressure (IOP) measurements obtained by Goldmann tonometry and dynamic contour tonometry (n=451, R=0.853, p<0.001).
new method to measure IOP
Which of these has the highest Pearson coefficient?
Fig4: Correlation analysis of the EndoPredict test results in the seven different pathology laboratories. a–g Results of the individual laboratories. h Pearson correlation coefficients
R=0.999R=0.987
How to Create a Linear Regression Equation with Microsoft Excel
A scatter plot will show you where your points lie will give you a visual clue about whether your data is linear, exponential or some either type of relationship.
Therefore, if you aren’t sure your data is linear in nature, create a scatter plot.
Finding a linear regression equation via a scatter plot and a trendline.
Figure 1: Mass versus volume for metal objects
y = 3.3534x - 1.562
R2 = 0.9842
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Volume (mL)
Mass (g)
If you know that one variable causes the changes in the other variable, then you can use linear regression to investigate the relation. This fits a straight line to the data, and gives the values of the slope and intercept ofthat line (m and b in the equation y = mx + b).
The simplest way to do this in Excel is to plot a scatter graph of the data and use the trend line feature of the graph.Right-click on a data point on the graph, select Add Trend line, and choose Linear.Click on the Options tab, and select Display equation on chart. You can also choose to set the intercept to be zero (or some other value). The full equation with the slope and intercept values are now shown on the chart.
Step 1: Enter your data into an EXCEL file
Left column x, right column is y
Step 2: Create a scatter plot for your dataINSERT / Chart / select XY(scatter) in chart wizard
Volume (mL) mass (g)4.2 12.56.3 18.63.5 10.52.4 6.66.9 22.75.2 15.3
Step 2: Create a scatter plot for your dataINSERT / Chart / select XY(scatter) in chart wizard
Step 3: Click anywhere on the graph.Step 4: Click the “Chart” tab and then chart options to modify things on the graph
Make these lines bigger
Step 5: Click anywhere on the graph. Step 6: Click the “Chart” tab and then “add trendline”
Step 7: In the add trendline menu click the option button. Step 8: Click on the boxes… Set intercept = 0 (If you want line to include 0,0) Display equation on chart Display R-squared value on chart
Figure 1: Mass versus volume for metal objects
y = 3.3534x - 1.562
R2 = 0.9842
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Volume (mL)
Mass (g)
Volume (mL) mass (g)4.2 12.56.3 18.63.5 10.52.4 6.66.9 22.75.2 15.3
You can move this
and add a white background
slope = 3.35 gmL
The R2 value is close to +1… what does this mean?What is the Pearson correlation constant?
r = R2 = 0.9842 = 0.992
Figure 1. Mass versus volume
y = 3.0566x
R2 = 0.9756
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Mass (g)
This is the same graph with y intercept set to 0
slope = 3.06 gmL
Why should it pass 0,0 ?
Make these lines bigger
Volume (mL) +/- .2mL mass (g) +/- 0.5g4.2 12.56.3 18.63.5 10.52.4 6.66.9 22.75.2 15.3
Figure 1: Mass versus volume for metal objects
y = 3.3534x - 1.562
R2 = 0.9842
0
5
10
15
20
25
0 2 4 6 8
Volume (mL)
Mass (g)
Volume (mL) +/- .2mL mass (g) +/- 0.5g4.2 12.56.3 18.63.5 10.52.4 6.66.9 22.75.2 15.3
Figure 1: Mass versus volume for metal objects
y = 3.3534x - 1.562
R2 = 0.9842
0
5
10
15
20
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0 2 4 6 8
Volume (mL)
Mass (g)
Click on data points
Volume (mL) +/- .2mL mass (g) +/- 0.5g4.2 12.56.3 18.63.5 10.52.4 6.66.9 22.75.2 15.3
0.5
Figure 1: Mass versus volume for metal objects
y = 3.3534x - 1.562
R2 = 0.9842
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5
10
15
20
25
0 2 4 6 8
Volume (mL)
Mass (g)
In Excel r is calculated using the formula: = CORREL (X range, Y range) .
pair female male1 17.1 16.52 18.5 17.43 19.7 17.34 16.2 16.85 21.3 19.56 19.6 18.3r= 0.8813
Example: The size of breeding pairs of penguins was measured to see if there was correlation between the sizes of the two sexes.
It is usual to draw a scatter graph of the data whenever a correlation is being investigated.
Insert | Function | CORREL
The scatter graph and both correlation coefficients clearly indicate a strong positive correlation.
Figure 1: Corelation of female and male penguins
y = 0.5205x + 7.883
R2 = 0.7767
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male penguin mass (kg)
female penguin mass (kg)
r = R2
r = 0.7767 = 0.8813
R can be calculated from R2
It is usual to draw a scatter graph of the data whenever a correlation is being investigated.
In other words large females do pair with large males. Of course this doesn't say why, but it shows there is a correlation to investigate further.
Figure 1: Corelation of female and male penguins
y = 0.5205x + 7.883
R2 = 0.7767
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male penguin mass (kg)
female penguin mass (kg)
Causation and correlation ?1.1.6 Explain that the existence of a correlation does not
establish that there is a causal relationship between two variables..Typically in Biology your experiment may involve a continuous independent variable and a continuously variable dependent variable. e.g effect of enzyme concentration on the rate of an enzyme catalyzed reaction.
The statistical analysis would set out to test the strength of the relationship (correlation). Once a correlation between two factors has been established from experimental data it would be necessary to advance the research to determine what the causal relationship might be.
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It is important to realize that if the statistical analysis of data indicates a correlation between the independent and dependent variable this does not prove any causation.
Only further investigation will reveal the causal effect between the two variables.
Causation
Skirt lengths and stock prices are highly correlated (as stock prices go up, skirt lengths get shorter).
The number of cavities in elementary school children and vocabulary size have a strong positive correlation.
Clearly there is no real interaction between the factors involved simply a co-incidence of the data.
Correlation does not imply causation!
Correlation vs. Causation :We have been discussing correlation. We have looked at situations where there exists a strong positive relationship between our variables x and y. However, just because we see a strong relationship between two variables, this does not imply that a change in one variable causes a change in the other variable. Correlation does not imply causation! Consider the following:
In the 1990s, researchers found a strong positive relationship between the number of television sets per person x and the life expectancy y of the citizens in different countries. That is, countries with many TV sets had higher life expectancies. Does this imply causation? By increasing the number of TVs in a country, can we increase the life expectancy of their citizens? Are there any hidden variables that may explain this strong positive correlation?
There is a strong positive correlation between ice cream sales and shark attacks. That is, as ice cream sales increase, the number of shark attacks increase. Is it reasonable to conclude the following? Ice cream consumption causes shark attacks.
All of the previous examples show a strong positive correlation between the variables. However, in each example it is not the case that one variable causes a change in the other variable. For example, increasing the number of ice cream sales does not increase the number of shark attacks. There are outside factors, also known as lurking variables, which cause the correlation between these variables.
Correlation does not always mean that one thing causes the other thing (causation), because a something else might have caused both.
For example, on hot days people buy ice cream, and people also go to the beach where some are eaten by sharks. There is a correlation between ice cream sales and shark attacks (they both go up as the temperature goes up in this case). But just because ice cream sales go up does not cause (causation) more shark attacks.
Correlation does not imply causation!
You may be interested to know that global warming, earthquakes, hurricanes, and other natural disasters are a direct effect of the shrinking numbers of Pirates since the 1800s. For your interest, I have included a graph of the approximate number of pirates versus the average global temperature over the last 200 years. As you can see, there is a statistically significant inverse relationship between pirates and global temperature.
