This paper is concerned with the overall significance and reproducibility of the response function. A test of significance is devised which is based on the Binomial distribution. Combined with other tests, the method is then used to compare two different response functions to examine the reproducibility of climate-chronology response. Two approaches are used: the first compares two response functions covering the same period from the same site, based on independent chronologies of the same species; the second compares the response of a single chronology over two equal non-overlapping time periods. The results suggest that the response in the examples used is statistically reproducible on a site, and statistically stable over periods of time.

A new approach to removing the non-climatic variance of forest interior tree-ring width series, using the smoothing spline, is described. This method is superior to orthogonal polynomials because it makes no assumptions about the shape of the curve to be used for standardization. Also, because the spline curve can range continuously from a linear least squares fit to cubic interpolation through the data, it is far more flexible than polynomials and provides a more "natural" fit. For computing the spline, we found that specifying the Lagrange multiplier p which appears in the calculus of variation solution rather than the residual variance as suggested by Reinsche was both practical and more efficient. In effect, the smoothing spline is a one-parameter family of low-pass filters defined by p. We describe the general characteristics of these filters in the time and frequency domains and compute the response functions for several of them. The smoothing spline is an excellent tree-ring standardization method because its filtering characteristics are well defined. Its utility for dendroclimatology should be considerable since, outside of semiarid environments, sites similar to forest interiors predominate.

Tests of the compound increment function, introduced by Warren (1980) as a means for removing the growth trend from dendrochronological data, are herein reported. In particular, the inter- and intra-site correlations of the residuals generated by the new method are compared with those generated by standard exponential fits. It is also shown that, in the presence of non-climatically induced responses, such as might arise from thinning, exponential fits can lead to spuriously high intra-site correlations. Accordingly, and because the new method's virtual elimination of negative and very low positive correlations, it appears to be the more satisfactory for portraying the growth trend.

The average ring-width index of two published chronologies from the eastern Tennessee climatic division was used as a single predictor variable in linear regression to reconstruct May June precipitation. Regression equations obtained using regionally averaged precipitation data from stations within the climatic division were compared with regressions obtained using single-station (Knoxville, Tennessee) data for comparable periods. The (regionally averaged) division data always provided the better calibration statistics for the regression equations. When the regressions calibrated using division data were verified with independent data for the climatic division and for Knoxville, the better results were always obtained for the division data. When regressions calibrated using single-station data were verified, the independent division data once again provided better results than the independent Knoxville data. Regionally averaged precipitation data also provided more satisfactory results than single- station data in a similar experiment for central Iowa, and probably provides better results in general for this type of dendroclimatic experiment.

The use of orthogonalized climatic variables in regression to specify treegrowth/climate relationships, commonly known as response function analysis, involves several a priori decisions and a posteriori interpretations, any of which maybe open to question. Decisions about the number of climatic variables to include, confidence limits, number of eigenvectors to allow as candidate predictors in regression, etc., can affect the response function in unpredictable ways and lead to possible errors in interpretation. To demonstrate the nature of these effects, we compared response functions for particular chronologies with the correlation function, which is simply the series of correlation coefficients between a tree-ring chronology and each of several sequential monthly climatic variables. The results indicate that response functions including high-order eigenvectors should be interpreted cautiously, and we recommend using the correlation function as an interpretive guide. Prior tree-growth variables in regression can mask climatic effects, and the correlation function can also be useful in detecting this masking. Statistical significance is more often attained in response functions than in correlation functions, possibly due to differences in the statistical testing procedures, to the statistical efficiency of eigenvectors in spending degrees of freedom, or to the filtering effects on the climatic data that result from eliminating high-order eigenvectors (noise) from the response function. These filtering effects plus the orthogonalization make response function analysis an efficient method for specifying tree-growth/climate relationships. The examples and guidelines presented here should enhance the usefulness of the method.

This paper is concerned with examining the degree of correlation between monthly climatic variables (multicollinearity) within data sets selected for their high quality. Various methods of describing the degree of multicollinearity are discussed and subsequently applied to different combinations of climate data within each site. The results indicate that higher degrees of multicollinearity occur in shorter data sets. Data consisting of 12 monthly variables of a single parameter (temperature or precipitation) have very low degrees of multicollinearity. Data set combinations of two parameters and lagged variables, as commonly used in tree-ring response function analysis, can have significant degrees of multicollinearity. If no preventative or corrective measures are taken when using such multicollinear data, erroneous interpretations of regression results may occur.

Three different response-function programs are applied to three tree-ring chronologies, two of Pinus ponderosa Laws. and one of Juniperus occidentalis Hook. These data were analyzed before and after ARMA modeling was applied. The results are described and compared to one another as well as to those obtained from all-steps multiple regression, stepwise multiple regression, ridge regression, and simple correlation. In spite of methodological differences all multivariate methodsproduced remarkably similar results. The results from simple correlation differed the most. Some differences among the response functions were apparent, especially in the coefficients associated with prior growth. The response-function results have smaller error estimates than ridge regression. According to Cropper (1985) this error in the response-function results is underestimated by approximately 40 %. The rationale for the different response-function solutions is discussed.

I propose an approach to provide 95% confidence intervals for a chronology of low-frequency tree-ring variation so that a level of significance or importance for trends can be inferred. The approach also visually reveals the portions of a chronology in which sample depth is so poor that low-frequency variation is not robustly estimated. A key characteristic of the approach is that it is essentially a reordering of the individual steps commonly used in constructing standard tree-ring chronologies; consequently, it is computationally simple for researchers who already routinely construct standard tree-ring chronologies. The most important ramification of the approach is that each year of the chronology has a distribution of smoothed index values with which to estimate confidence intervals around the chronology of low-frequency variation. It can be argued that the approach constitutes multiple significance testing of means, which causes the α level for the confidence interval to be unknown. Nonetheless, the approach is still useful in that it provides a way to evaluate the probable importance of low-frequency trends expressed in tree-ring chronologies.

The bootstrap procedure provides a way to test the significance of the regression coefficients and the stability of the estimates in response functions generated by regression on principal components. A subroutine RESBO, which calculates a bootstrapped response function, has been added to Fritts' program PRECON.

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