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Understanding Your Data Set • Statistics are used to describe data sets • Gives us a metric in place of a graph • What are some types of statistics used to describe data sets? – Average, range, variance, standard deviation, coefficient of variation, standard error Table 1. Total length (cm) and average length of spotted gar collected from a local farm pond and from a local lake. Length Number Pond Lake 1 34 38 2 78 82 3 48 58 4 24 76 5 64 60 6 58 70 7 34 99 8 66 40 9 22 68 10 44 91 47.2 68.2 Average= Length Number Pond Lake 1 34 38 2 78 82 3 48 58 4 24 76 5 64 60 6 58 70 7 34 99 8 66 40 9 22 68 10 44 91 47.2 68.2 Average= • Are the two samples equal? – What about 47.2 and 47.3? • If we sampled all of the gar in each water body, would the average be different? – How different? • Would the lake fish average still be larger? Range • Simply the distance between the smallest and largest value Lake Overlap Pond 0 20 40 60 80 100 Length (cm) Figure 1. Range of spotted gar length collected from a pond and a lake. The dashed line represents the overlap in range. • Does the difference in average length (47.2 vs. 68.2) seem to be much as large as before? Lake Overlap Pond 0 20 40 Length (cm) 60 80 100 Variance • An index of variability used to describe the dispersion among the measures of a population sample. • Need the distance between each sample point and the sample mean. 100 Distance from point to the sample mean Length 80 60 40 20 0 0 2 4 6 8 10 Number Figure 2. Mean length (cm) of each spotted gar collected from the pond. The horizontal solid line represents the sample mean length. • We can easily put this new data set into a spreadsheet table. • By adding up all of the differences, we can get a number that is a reflection of how scattered the data points are. # Length Mean Difference 1 34 47.2 -13.2 2 78 47.2 30.8 3 48 47.2 0.8 4 24 47.2 -23.2 5 64 47.2 16.8 6 58 47.2 10.8 • After adding up all of the differences, we get zero. 7 34 47.2 -13.2 8 66 47.2 18.8 – This is true of all calculations like this 9 22 47.2 -25.2 10 44 47.2 -3.2 Sum = 0 – Closer to the mean each number is, the smaller the total difference. • What can we do to get rid of the negative values? Sum of Squares # Length Mean Difference Difference2 1 34 47.2 -13.2 174.24 2 78 47.2 30.8 948.64 3 48 47.2 0.8 0.64 4 24 47.2 -23.2 538.24 5 64 47.2 16.8 282.24 6 58 47.2 10.8 116.64 7 34 47.2 -13.2 174.24 8 66 47.2 18.8 353.44 9 22 47.2 -25.2 635.04 10 44 47.2 -3.2 10.24 0 3233.6 Sum = Now 3233.6 is a number we can use! This value is called the SUM OF SQUARES. Back to Variance • Sum of Squares (SOS) will continue to increase as we increase our sample size. – A sample of 10 replicates that are highly variable would have a higher SOS than a sample of 100 replicates that are not highly variable. • To account for sample size, we need to divide SOS by the number of samples minus one (n-1). – We’ll get to the reason (n-1) instead of n later Calculate Variance (σ2) σ2 = S2 = (Xi – Xm)2 / (n – 1) Degrees of Freedom SOS Variance for Pond = S2 = 3233.6 / 9 = 359.29 100 Distance from point to the sample mean Length 80 60 40 20 0 0 2 4 6 Number 8 10 More on Variance • Variance tends to increase as the sample mean increases – For our sample, the largest difference between any point and the mean was 30.8 cm. Imagine measuring a plot of cypress trees. How large of a difference would you expect (if measured in cm)? • The variance for the lake sample = 400.18. Standard Deviation • Calculated as the square root of the variance. – Variance is not a linear distance (we had to square it). Think about the difference in shape of a meter stick versus a square meter. • By taking the square root of the variance, we return our index of variability to something that can be placed on a number line. Calculate SD • For our gar sample, the Variance was 359.29. The square root of 359.29 = 18.95. – Reported with the mean as: 47.2 ± 18.95 (mean ± SD). • Standard Deviation is often abbreviated as σ (sigma) or as SD. • SD is a unit of measurement that describes the scatter of our data set. – Also increases with the mean Standard Error • Calculated as: SE = σ / √(n) – Indicates how close we are to estimating the true population mean – For our pond ex: SE = 18.95 / √10 = 5.993 – Reported with the mean as 47.2 ± 5.993 (mean ± SE). – Based on the formula, the SE decreases as sample size increases. • Why is this not a mathematical artifact, but a true reflection of the population we are studying? Sample Size • The number of individuals within a population you measure/observe. – Usually impossible to measure the entire population • As sample size increases, we get closer to the true population mean. – Remember, when we take a sample we assume it is representative of the population. Effect of Increasing Sample Size • I measured the length of 100 gar • Calculated SD and SE for the first 10, then included the next additional 10, and so on until all 100 individuals were included. Raw Data 120 100 80 60 40 20 0 0 20 40 60 Sample Size 80 100 120 SD = Square root of the variance (Var = (Xi – Xm) / (n – 1)) SD 90 80 70 60 50 40 0 20 40 60 Sample Size 80 100 SE = SD / √(n) SE 90 80 70 60 50 40 0 20 40 60 Sample Size 80 100 SD 24 22 20 18 16 14 12 0 20 40 60 80 100 60 80 100 SE 10 8 6 4 2 0 0 20 40 Population: a data set representing the entire entity of interest - What is a population? Sample: a data set representing a portion of a population Population Sample Population mean – the true mean for that population -a single number Sample mean – the estimated population mean -a range of values (estimate ± 95% confidence interval) Population Sample As our sample size increases, we sample more and more of the population. Eventually, we will have sampled the entire population and our sample distribution will be the population distribution Increasing sample size Ni=x Mean = x = N (x-x)2 Variance = N-1 Individual 1 2 3 4 5 6 N=6 N-1=5 Weight 26 32 25 26 30 30 169 Standard Deviation = (x-x)2 N-1 SD Standard Error = 2 Mean (Weight - Mean) 28.17 4.7089 28.17 14.6689 28.17 10.0489 28.17 4.7089 28.17 3.3489 28.17 3.3489 SOS= 40.8334 √N Mean = 169/6 = 28.17 Range = 25 – 32 SOS = 40.83 Variance = 40.83 / 5 = 8.16 Std. Dev. = 40.83/5 = 2.86 Std. Err. = 2.86 / √6 = 1.17 Go to Excel MEAN ± CONFIDENCE INTERVAL When a population is sampled, a mean value is determined and serves as the point-estimate for that population. However, we cannot expect our estimate to be the exact mean value for the population. Instead of relying on a single point-estimate, we estimate a range of values, centered around the point-estimate, that probably includes the true population mean. That range of values is called the confidence interval. Confidence Interval Confidence Interval: consists of two numbers (high and low) computed from a sample that identifies the range for an interval estimate of a parameter. There is a 5% chance (95% confidence interval) that our interval does not include the true population mean. y ± (t/0.05)[() / (n)] 28.17 ± 2.29 25.88 30.45 •Hypothesis Testing –Null versus Alternative Hypothesis •Briefly: –Null Hypothesis: Two means are not different –Alternative Hypothesis: Two means are not similar •A test statistic based on a predetermined probability (usually 0.05) is used to reject or accept the null hypothesis < 0.05 then there is a significant difference > 0.05 then there is NO significant difference Are Two Populations The Same? • Boudreaux: ‘My pond is better than your lake, cher’! • Alphonse: ‘Mais non! I’ve got much bigger fish in my lake’! • How can the truth be determined? Two Sample t-test • Simple comparison of a specific attribute between two populations • If the attributes between the two populations are equal, then the difference between the two should be zero • This is the underlying principle of a t-test • If P-value > 0.05 the means are not significantly different; If P < 0.05 the means are significantly different Analysis of Variance Can compare two or more means • Compares means to determine if the population distributions are not similar • Uses means and confidence intervals much like a t-test • Test statistic used is called an F statistic (F-test), which is used to get the P value • If P-value > 0.05 the means are not significantly different; If P< 0.05 the means are significantly different • Post-hoc test separates the non-similar ones Analysis of Variance • Compares means to determine if the population distributions are not similar • Uses means and confidence intervals much like a t-test • Test statistic used is called an F statistic (F-test) Normal Distribution • Most characteristics follow a normal distribution – For example: height, length, speed, etc. • One of the assumptions of the ANOVA test is that the sample data is ‘normally distributed.’ Sample Distribution Approaches Normal Distribution With Sample Size 10 Frequency 8 6 4 2 0 Population Sample Sample Distribution Approaches Normal Distribution With Sample Size 10 Frequency 8 6 4 2 0 Population Sample Sample Distribution Approaches Normal Distribution With Sample Size 10 Frequency 8 6 4 2 0 Population Sample Mean = x = Variance = Individual 1 2 3 4 5 6 N=6 N-1=5 Weight 26 32 25 26 30 30 169 Ni=x N (x-x)2 N-1 Standard Deviation = (x-x)2 N-1 SD Standard Error = 2 Mean (Weight - Mean) 28.17 4.7089 28.17 14.6689 28.17 10.0489 28.17 4.7089 28.17 3.3489 28.17 3.3489 SOS= 40.8334 √N Mean = 169/6 = 28.17 Range = 25 – 32 SOS = 40.83 Variance = 40.83 / 5 = 8.16 Std. Dev. = 40.83/5 = 2.86 Std. Err. = 2.86 / √6 = 1.17 ANOVA – Analysis of Variance Calculate a SOS based on an overall mean (total SOS) Pond Lake 120 100 80 60 40 20 0 0 1 2 3 Trtmnt Replicate Length Overall Mean SOSTotal Pond 1 34 57.7 561.69 Pond 2 78 57.7 412.09 Pond 3 48 57.7 94.09 Pond 4 24 57.7 1135.69 Pond 5 64 57.7 39.69 Pond 6 58 57.7 0.09 Pond 7 34 57.7 561.69 Pond 8 66 57.7 68.89 Pond 9 22 57.7 1274.49 Pond 10 44 57.7 187.69 80 Lake 1 38 57.7 388.09 60 Lake 2 82 57.7 590.49 Lake 3 58 57.7 0.09 Lake 4 76 57.7 334.89 Lake 5 60 57.7 5.29 Lake 6 70 57.7 151.29 Lake 7 99 57.7 1705.69 Lake 8 40 57.7 313.29 Lake 9 68 57.7 106.09 Lake 10 91 57.7 1108.89 This provides a measure of the overall variance (Total SOS). Pond Lake 120 9040.2 100 40 20 0 0 1 2 3 Calculate a SOS based for each treatment (Treatment or Error SOS). Pond Lake 120 100 80 60 40 20 0 0 1 2 3 Trtmnt Replicate Length Trtmnt Mean SOSError Pond 1 34 47.2 174.24 Pond 2 78 47.2 948.64 Pond 3 48 47.2 0.64 Pond 4 24 47.2 538.24 Pond 5 64 47.2 282.24 Pond 6 58 47.2 116.64 Pond 7 34 47.2 174.24 Pond 8 66 47.2 353.44 Pond 9 22 47.2 635.04 Pond 10 44 47.2 10.24 Lake 1 38 68.2 912.04 Lake 2 82 68.2 190.44 Lake 3 58 68.2 104.04 Lake 4 76 68.2 60.84 Lake 5 60 68.2 67.24 Lake 6 70 68.2 3.24 Lake 7 99 68.2 948.64 Lake 8 40 68.2 795.24 60 Lake 9 68 68.2 0.04 20 Lake 10 91 68.2 519.84 This provides a measure of the reduction of variance by measuring each treatment separately (Treatment or Error SOS). What happens to Error SOS when the variability w/in each treatment decreases? Pond Lake 120 100 80 40 0 6835.2 0 1 2 3 Calculate a SOS for each predicted value vs. the overall mean (Model SOS) Predicted_Pond Predicted_Lake Overall_Avg 120 100 80 60 40 20 0 0 1 2 3 Trtmnt Replicate Length Trtmnt Mean Overall Mean SOSModel Pond 1 34 47.2 57.7 110.25 Pond 2 78 47.2 57.7 110.25 Pond 3 48 47.2 57.7 110.25 Pond 4 24 47.2 57.7 110.25 Pond 5 64 47.2 57.7 110.25 Pond 6 58 47.2 57.7 110.25 Pond 7 34 47.2 57.7 110.25 Pond 8 66 47.2 57.7 110.25 Pond 9 22 47.2 57.7 110.25 Pond 10 44 47.2 57.7 110.25 Lake 1 38 68.2 57.7 110.25 Lake 2 82 68.2 57.7 110.25 Lake 3 58 68.2 57.7 110.25 Lake 4 76 68.2 57.7 110.25 Lake 5 60 68.2 57.7 110.25 Lake 6 70 68.2 57.7 110.25 Lake 7 99 68.2 57.7 110.25 Lake 8 40 68.2 57.7 110.25 Lake 9 68 68.2 57.7 110.25 Lake 10 91 68.2 57.7 110.25 2205 This provides a measure of the distance between the mean values (Model SOS). What happens to Model SOS when the two means are close together? What if the means are equal? Detecting a Difference Between Treatments • Model SOS gives us an index on how far apart the two means are from each other. – Bigger Model SOS = farther apart • Error SOS gives us an index of how scattered the data is for each treatment. – More variability = larger Error SOS = more possible overlap between treatments Magic of the F-test • The ratio of Model SOS to Error SOS (Model SOS divided by Error SOS) gives us an overall index (the F statistic) used to indicate the relative ‘distance’ and ‘overlap’ between two means. – A large Model SOS and small Error SOS = a large F statistic. Why does this indicate a significant difference? – A small Model SOS and a large Error SOS = a small F statistic. Why does this indicate no significant difference?? • Based on sample size and alpha level (P-value), each F statistic has an associated P-value. – P < 0.05 (Large F statistic) there is a significant difference between the means – P ≥ 0.05 (Small F statistic) there is NO significant difference Showing Results 35 30 A A 25 B 20 15 10 5 0 1 2 3 Regression • For the purposes of this class: – Does Y depend on X? – Does a change in X cause a change in Y? – Can Y be predicted from X? • Y= mX + b Predicted values 180 Dependent Value Actual values 160 140 120 Overall Mean 100 30 40 50 60 Independent Value 70 80 When analyzing a regression-type data set, the first step is to plot the data: Y 114 120 150 140 166 138 180 Dependent Value (Y) X 35 45 55 65 75 55 160 140 120 100 30 40 50 60 70 Independent Value (X) The next step is to determine the line that ‘best fits’ these points. It appears this line would be sloped upward and linear (straight). 80 The line of best fit is the sample regression of Y on X, and its position is fixed by two results: Y = 1.24(X) + 69.8 Dependent Value 180 slope 160 Y-intercept 140 (55, 138) 120 Rise/Run 100 30 40 50 60 70 80 Independent Value 1) The regression line passes through the point (Xavg, Yavg). 2) Its slope is at the rate of “m” units of Y per unit of X, where m = regression coefficient (slope; y=mx+b) Testing the Regression Line for Significance • An F-test is used based on Model, Error, and Total SOS. – Very similar to ANOVA • Basically, we are testing if the regression line has a significantly different slope than a line formed by using just Y_avg. – If there is no difference, then that means that Y does not change as X changes (stays around the average value) • To begin, we must first find the regression line that has the smallest Error SOS. Error SOS The regression line should pass through the overall average with a slope that has the smallest Error SOS (Error SOS = the distance between each point and predicted line: gives an index of the variability of the data points around the predicted line). Dependent Value 180 160 140 138 overall average is the pivot point 120 100 30 40 50 55 60 Independent Value 70 80 For each X, we can predict Y: Y = 1.24(X) + 69.8 Error SOS is calculated as the sum of (YActual – YPredicted)2 This gives us an index of how scattered the actual observations are around the predicted line. The more scattered the points, the larger the Error SOS will be. This is like analysis of variance, except we are using the predicted line instead of the mean value. SOSErro X Y_Actual Y_Pred r 35 114 113.2 0.64 45 120 125.6 31.36 55 150 138 144 65 140 150.4 108.16 75 166 162.8 10.24 294.4 Total SOS • Calculated as the sum of (Y – Yavg)2 • Gives us an index of how scattered our data set is around the overall Y average. Dependent Value 180 Regression line not shown 160 140 Overall Y average 120 100 30 40 50 60 Independent Value 70 80 Total SOS gives us an index of how scattered the data points are around the overall average. This is calculated the same way for a single treatment in ANOVA. X Y_Actual Y Average 35 114 138 576 45 120 138 324 55 150 138 144 65 140 138 4 75 166 138 784 SOSTotal 1832 What happens to Total SOS when all of the points are close to the overall average? What happens when the points form a non-horizontal linear trend? Model SOS • Calculated as the Sum of (YPredicted – Yavg)2 • Gives us an index of how far all of the predicted values are from the overall average. Dependent Value 180 Distance between predicted Y and overall mean 160 140 120 100 30 40 50 60 Independent Value 70 80 Model SOS • Gives us an index of how far away the predicted values are from the overall average value Y Avera ge SOSMod X Y_Pred 35 113.2 138 615.04 45 125.6 138 153.76 55 138 138 0 65 150.4 138 153.76 75 162.8 138 615.04 el • What happens to Model SOS when 1537.6 all of the predicted values are close to the average value? All Together Now!! X Y_Actual Y_Pred SOSError Y_Avg SOSTotal SOSModel 35 114 113.2 0.64 138 576 615.04 45 120 125.6 31.36 138 324 153.76 55 150 138 144 138 144 0 65 140 150.4 108.16 138 4 153.76 75 166 162.8 10.24 138 784 615.04 1832 1537.6 294.4 SOSError = (Y_Actual – Y_Pred)2 SOSTotal = (Y_Actual –Y_ Avg) 2 SOSMode l = (Y_Pred – Y_Avg) 2 Using SOS to Assess Regression Line • Model SOS gives us an index on how ‘different’ the predicted values are from the average values. – Bigger Model SOS = more different – Tells us how different a sloped line is from a line made up only of Y_avg. – Remember, the regression line will pass through the overall average point. • Error SOS gives us an index of how different the predicted values are from the actual values – More variability = larger Error SOS = large distance between predicted and actual values Magic of the F-test • The ratio of Model SOS to Error SOS (Model SOS divided by Error SOS) gives us an overall index (the F statistic) used to indicate the relative ‘difference’ between the regression line and a line with slope of zero (all values = Y_avg. – A large Model SOS and small Error SOS = a large F statistic. Why does this indicate a significant difference? – A small Model SOS and a large Error SOS = a small F statistic. Why does this indicate no significant difference?? • Based on sample size and alpha level (P-value), each F statistic has an associated P-value. – P < 0.05 (Large F statistic) there is a significant difference between the regression line a the Y_avg line. – P ≥ 0.05 (Small F statistic) there is NO significant difference between the regression line a the Y_avg line. Mean Model SOS = F Mean Error SOS Dependent Value 180 160 140 120 100 40 50 60 70 80 Independent Value 180 Dependent Value Basically, this is an index that tells us how different the regression line is from Y_avg, and the scatter of the data around the predicted values. 30 160 140 120 100 30 40 50 60 70 Independent Value 80 Correlation (r): Another measure of the mutual linear relationship between two variables. • ‘r’ is a pure number without units or dimensions • ‘r’ is always between –1 and 1 • Positive values indicate that y increases when x does and negative values indicate that y decreases when x increases. – What does r = 0 mean? • ‘r’ is a measure of intensity of association observed between x and y. – ‘r’ does not predict – only describes associations between variables Dependent Variable 180 Dependent Variable 180 r>0 160 140 160 140 r<0 120 100 120 30 40 50 60 70 Independent Variable 100 30 40 50 60 70 80 Inpendent Variable Dependent Variable 180 r is also called Pearson’s correlation coefficient. r=0 160 140 120 100 30 40 50 60 Independent Variable 70 80 80 R-square • If we square r, we get rid of the negative value if it is negative) and we get an index of how close the data points are to the regression line. • Allows us to decide how much confidence we have in making a prediction based on our model. • Is calculated as Model SOS / Total SOS r2 = Model SOS / Total SOS = Model SOS 180 Dependent Value = Total SOS 160 140 120 100 30 40 50 60 Independent Value 70 80 = Model SOS 180 Dependent Value = Total SOS R2 = 0.8393 160 r2 = Model SOS / Total SOS 140 numerator/denominator 120 100 30 40 50 60 70 80 Independent Value 1.2 R2 = 0.0144 1 0.8 Small numerator Big denominator 0.6 0.4 0.2 0 0 10 20 30 40 50 R-square and Prediction Confidence 1.2 1.2 2 R = 0.0144 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 0 60 1.2 R2 = 0.5537 10 20 30 40 50 60 40 50 60 1.2 1 1 R2 = 0.7605 R2 = 0.9683 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 10 20 30 40 50 60 0 10 20 30 Finally…….. • If we have a significant relationship (based on the p-value), we can use the r-square value to judge how sure we are in making a prediction.