Simple Regression - Y1 vs. t
Dependent variable: Y1
Independent variable: t
Logarithmic-Y squared-X: Y = exp(a + b*X^2)
Number of observations: 20

Coefficients
| Least Squares | Standard | T |
Parameter | Estimate | Error | Statistic | P-Value
Intercept | 9,18851 | 0,0326876 | 281,1 | 0,0000
Slope | 0,00122767 | 0,000171961 | 7,13926 | 0,0000

NOTE: intercept = ln(a)

Analysis of Variance
Source | Sum of Squares | Df | Mean Square | F-Ratio | P-Value
Model | 0,468464 | 1 | 0,468464 | 50,97 | 0,0000
Residual | 0,165441 | 18 | 0,00919115 | |
Total (Corr.) | 0,633905 | 19 | | |

Correlation Coefficient = 0,859659
R-squared = 73,9013 percent
R-squared (adjusted for d.f.) = 72,4514 percent
Standard Error of Est. = 0,0958705
Mean absolute error = 0,0688858
Durbin-Watson statistic = 0,825561 (P=0,0006)
Lag 1 residual autocorrelation = 0,520078

The StatAdvisor
The output shows the results of fitting a logarithmic-Y squared-X model to describe the relationship between Y1 and t.  The equation of the fitted model is

   Y1 = exp(9,18851 + 0,00122767*t^2)

Since the P-value in the ANOVA table is less than 0,05, there is a statistically significant relationship between Y1 and t at the 95,0% confidence level.

The R-Squared statistic indicates that the model as fitted explains 73,9013% of the variability in Y1.  The correlation coefficient equals 0,859659, indicating a moderately strong relationship between the variables.  The standard error of the estimate shows the standard deviation of the residuals to be 0,0958705.  This value can be used to construct prediction limits for new observations by selecting the Forecasts option from the text menu.

The mean absolute error (MAE) of 0,0688858 is the average value of the residuals.  The Durbin-Watson (DW) statistic tests the residuals to determine if there is any significant correlation based on the order in which they occur in your data file.  Since the P-value is less than 0,05, there is an indication of possible serial correlation at the 95,0% confidence level.  Plot the residuals versus row order to see if there is any pattern that can be seen.  

The output shows the results of fitting a logarithmic-Y squared-X model to describe the relationship between Y1 and t.  The equation of the fitted model, shown as a solid line, is

   Y1 = exp(9,18851 + 0,00122767*t^2)

The inner bounds show 95,0% confidence limits for the mean Y1 of many observations at given values of t.  The outer bounds show 95,0% prediction limits for new observations.  

Predicted Values
| Predicted | Lower 95% | Upper 95% | Lower 95% | Upper 95%
X | Y | Pred. Limit | Pred. Limit | Conf. Limit | Conf. Limit
21,0 | 16813,1 | 13322,6 | 21218,2 | 14963,7 | 18891,2
22,0 | 17724,5 | 13938,8 | 22538,4 | 15548,3 | 20205,4
23,0 | 18731,3 | 14602,7 | 24027,0 | 16180,7 | 21683,8

The StatAdvisor
This table shows the predicted values for Y1 using the fitted model.  In addition to the best predictions, the table shows:

   (1) 95,0% prediction intervals for new observations   (2) 95,0% confidence intervals for the mean of many observations

The prediction and confidence intervals correspond to the inner and outer bounds on the graph of the fitted model.

This plot displays the Studentized residuals versus values of t.  Any non-random pattern could indicate that the selected model does not adequately describe the observed data.  In addition, any values outside the range of -3 to +3 could well be outliers.

Comparison of Alternative Models
Model | Correlation | R-Squared
Logarithmic-Y squared-X | 0,8597 | 73,90%
Reciprocal-Y squared-X | -0,8576 | 73,55%
Square root-Y squared-X | 0,8568 | 73,41%
Squared-X | 0,8516 | 72,52%
Double squared | 0,8350 | 69,72%
Reciprocal-Y | -0,8238 | 67,86%
Exponential | 0,8070 | 65,12%
Square root-Y | 0,7953 | 63,26%
Linear | 0,7819 | 61,14%
Logarithmic-Y square root-X | 0,7612 | 57,94%
Squared-Y | 0,7511 | 56,41%
Double square root | 0,7444 | 55,41%
Square root-X | 0,7262 | 52,73%
Multiplicative | 0,6920 | 47,89%
Squared-Y square root-X | 0,6874 | 47,25%
Square root-Y logarithmic-X | 0,6704 | 44,94%
Logarithmic-X | 0,6480 | 41,99%
Squared-Y logarithmic-X | 0,6025 | 36,30%
Double reciprocal | 0,5482 | 30,06%
S-curve model | -0,5000 | 25,00%
Reciprocal-X | -0,4506 | 20,31%
Squared-Y reciprocal-X | -0,4032 | 16,26%
Reciprocal-Y square root-X | <no fit> |
Reciprocal-Y logarithmic-X | <no fit> |
Square root-Y reciprocal-X | <no fit> |
Logistic | <no fit> |
Log probit | <no fit> |

The StatAdvisor
This table shows the results of fitting several curvilinear models to the data.  Of the models fitted, the logarithmic-Y squared-X model yields the highest R-Squared value with 73,9013%.  This is the currently selected model.

Unusual Residuals
| | | Predicted | | Studentized
Row | X | Y | Y | Residual | Residual
14 | 14,0 | 9845,0 | 12445,7 | -2600,74 | -3,04

The StatAdvisor
The table of unusual residualds lists all observations which have Studentized residuals greater than 2 in absolute value.  Studentize residuals measure how many standard deviations each observed value of Y1 deviates from a model fitted using all of the data except that observation.  In this case, there is one Studentized residual greater than 3.  You should take a careful look at the observations greater than 3 to determine whether they are outliers which should be removed from the model and handled separately.