
EQUAL SLOPES TESTName:
\( y_2 = a_2 + b_2 x_2 \) we want to test whether \( b_1 = b_2 \). For example, this might be of interest when we are fitting a regression line based on two different measurement methods and we would like to know if the fits are equivalent. If the residul variances from the two regressions are statistically equivalent, the test statstic is
where
\( c_2 = \frac{1}{Q_{x_1}} + \frac{1} {Q_{x_2}} \)
\( Q(x) = \sum{(x_{i}  \bar{x})^{2}} \) This test statistic is compared to a t distribution with n_{1} + n_{2}  4 degrees of freedom. If the residual variances are not statistically equivalent and n_{1} and n_{2} are both greater than 20, the test statistic is
This is compared to a standard normal distribution. If n_{1} or n_{2} is less than or equal to 20, the test statistic is compared to a t distribution with ν degrees of freedom where
\( c = \frac{\frac{s_{y_1 \cdot x_1}^{2}} {Q_{x_1}}} {\sqrt{ \frac{s_{y_1 \cdot x_1}^{2}} {Q_{x_1}} + \frac{s_{y_2 \cdot x_2}^{2}} {Q_{x_2}}} } \) Note that n_{1} should be set to the smaller sample size. To determine whether the residual variances are equal, the test statistic is
The hypothesis of equal residual variances is rejected if this statistic is greater than the F percent point function with n_{1} 2 and n_{2}  2 degrees of freedom. Dataplot will perform the test for equal residual variances first and apply the appropriate test based on this. For the case where three or more regression lines are being compared, a series of three tests are performed.
<SUBSET/EXCEPT/FOR qualification> where <y> is a response (= dependent) variable; <x> is a factor (= independent) variable; <tag> is a groupid variable; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
EQUAL SLOPE TEST Y X TAG SUBSET TAG 1 2 4 6
LET A = EQUAL SLOPES TEST CDF Y X TAG LET A = EQUAL SLOPES TEST PVALUE Y X TAG LET A = EQUAL SLOPES TEST CRITICAL VALUE Y X TAG The critical value is the 95% critical value. For the more than two groups case, the values returned are the second of the three tests (i.e., the test for equal slopes). In addition to the above LET command, builtin statistics are supported for 20+ different commands (enter HELP STATISTICS for details).
. Step 1: Read the data . dimension 40 columns read y x indx 0.04101 297.16000 0.00000 0.04104 297.17000 0.00000 0.04105 297.18000 0.00000 0.04103 297.20000 0.00000 0.04109 297.20000 0.00000 0.03707 280.12000 0.00000 0.03929 290.12000 0.00000 0.04171 300.18000 0.00000 0.04428 310.18000 0.00000 0.04700 320.17000 0.00000 0.04052 297.15000 1.00000 0.04055 297.15000 1.00000 0.04056 297.15000 1.00000 0.04056 297.15000 1.00000 0.04056 297.15000 1.00000 0.03656 280.15000 1.00000 0.03883 290.15000 1.00000 0.04125 300.15000 1.00000 0.04381 310.15000 1.00000 0.04654 320.15000 1.00000 end of data let n = size y . . Step 2: Generate the individual fits . set write decimals 6 write "Fit for Lab One" write " " write " " fit y x subset indx = 0 let pred1 = pred write " " write " " write "Fit for NIST" write " " write " " fit y x subset indx = 1 let pred2 = pred . . Step 3: Plot the fit . let x1 = x let x2 = x retain pred1 x1 subset indx = 0 retain pred2 x2 subset indx = 1 . case asis label case asis title case asis title offset 2 line blank blank solid solid line color black black blue red character circle box character fill on on character color blue red char hw 1.0 0.75 all y1label Y x1label X x2label Red: Lab 1, Blue: Lab 2 title Summary of Fits . ylimits 0.035 0.050 xlimits 280 320 xtic mark offset 5 5 . plot y x subset indx = 1 and plot y x subset indx = 0 and plot pred2 x2 and plot pred1 x1 . . Step 4: Perform the equal slopes test . let statva = equal slopes test y x indx let statcd = equal slopes test cdf y x indx let pval = equal slopes test pvalue y x indx let statcv = equal slopes test critical value y x indx . print statva statcd statcv pval . equal slopes test y x indxThe following output is generated Fit for Lab One Least Squares Multilinear Fit Sample Size: 10 Number of Variables: 1 Residual Standard Deviation: 0.000122 Residual Degrees of Freedom: 8 BIC: 177.777585 Replication Case: Replication Standard Deviation: 0.000042 Replication Degrees of Freedom: 1 Number of Distinct Subsets: 9 Lack of Fit F Ratio: 9.381749 Lack of Fit F CDF (%): 75.360484 Lack of Fit Degrees of Freedom 1: 7 Lack of Fit Degrees of Freedom 2: 1  Approximate Parameter Estimates Standard Deviation tValue  1 A0 0.032834 0.001143 28.7237 2 A1 X 0.000249 0.000004 65.0287 Fit for NIST Least Squares Multilinear Fit Sample Size: 10 Number of Variables: 1 Residual Standard Deviation: 0.000115 Residual Degrees of Freedom: 8 BIC: 179.031014 Replication Case: Replication Standard Deviation: 0.000017 Replication Degrees of Freedom: 4 Number of Distinct Subsets: 6 Lack of Fit F Ratio: 87.226813 Lack of Fit F CDF (%): 99.961750 Lack of Fit Degrees of Freedom 1: 4 Lack of Fit Degrees of Freedom 2: 4  Approximate Parameter Estimates Standard Deviation tValue  1 A0 0.033728 0.001075 31.3736 2 A1 X 0.000250 0.000004 69.5272 PARAMETERS AND CONSTANTS STATVA  0.265061 STATCD  0.397174 STATCV  2.119905 PVAL  0.794347 Summary Table  Sample Residual GroupID Size Intercept Slope Variance  1 10 0.032834 0.000249 0.000000 2 10 0.033728 0.000250 0.000000 Equal Slopes Test for Two Groups (Equal Residual Variances Case) Dependent (Y) Variable: Y Independent (X) Variable: X GroupID Variable: INDX H0: The Regression Slopes are Equal Ha: The Regression Slopes are not Equal Total Number of Observations: 20 Number of Groups with Ni > 3: 2 Test: Equal Slopes Test Statistic Value: 0.265061 CDF Value: 0.397174 PValue: 0.794347 Conclusions (TwoTailed tTest) H0: Slopes Are Equal  Null Significance Test Critical Hypothesis Level Statistic Region (+/) Conclusion  80.0% 0.265061 1.336757 ACCEPT 90.0% 0.265061 1.745883 ACCEPT 95.0% 0.265061 2.119905 ACCEPT 99.0% 0.265061 2.920773 ACCEPT  
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Date created: 07/21/2017 