We now illustrate the application of the above theory to the Irish daily rainfall record. Models were fitted sequentially, starting with “obvious” predictors and successively adding extra predictors and interactions. The value of adding successive predictors was assessed by examining the nominal log likelihood, predictive performance and residuals (see section 4.1 below) for each model. Initially, basic models corresponding to a stationary climate were fitted. To examine the evidence for changing rainfall patterns, these basic models were then augmented by adding predictors representing trends, together with their interactions.
 In selecting predictors to represent trends over time, four basic alternatives have been considered here. The first three are deterministic functions of time corresponding to trends that are linear, stationary before time t0 and then linear, and cyclical, respectively. Although it is implausible to extrapolate the first of these indefinitely outside the range of the data, it may well provide a good approximation to any monotonic trend over the period of record. The second is intended as a crude representation of anthropogenic climate change (t0 being the year in which the change started to occur). The cyclical trend was suggested by the Birr and Sligo records (see Figure 2).
 These trends are all essentially descriptive in nature. It is natural to ask whether there is a physical explanation for changing rainfall patterns, and to this end we have investigated the impact of the North Atlantic Oscillation (NAO) in addition to the deterministic trends. The NAO is known to be associated with European precipitation patterns, and its evolution since 1940 is not dissimilar to that of the winter rainfalls in Figure 2 [Hurrell, 1995]. The NAO index used in this study is the normalized monthly pressure difference between stations in Iceland and Gibraltar, defined by Jones et al. .
 Table 1 gives the number of parameters, nominal log likelihoods and root mean squared errors (RMSEs) for models incorporating various different trend scenarios. For the occurrence models the RMSE is defined as
where yi takes the value 1 if the ith case in the data set is a wet day and zero otherwise, and pi is the probability of rain under the model. As an error measure for binary data, this may be difficult to interpret; however, it is the square root of the mean Brier score which is commonly used for the evaluation of probability forecasts [Dawid, 1986].
 The log likelihoods clearly distinguish between the different models, and indicate that the best fits are obtained by occurrence model 6 and amounts model 8. For both occurrence and amounts, the NAO emerges as dominant among the trend scenarios considered. However, it does not account for all the trends in the data, since the likelihoods for occurrence model 5 and amounts model 6 are both significantly increased by adding extra terms corresponding to linear and cyclical trends respectively. For example, the nominal log likelihood for occurrence models 5 and 6 differ by 112.92; model 6 contains 8 additional parameters. If all sites were independent, a likelihood ratio test would compare 2 × 112.92 = 225.84 to tables of a χ2 distribution with 8 degrees of freedom; the p-value for the test would be 0.000 to 3 decimal places. Under complete dependence (see section 3.3), since there are 8.47 observations per day on average we would refer 225.84/8.47 = 26.66 to tables of the same distribution and obtain a p-value of 0.001. Hence there is strong evidence that model 6 improves upon model 5, even after accounting for intersite dependence. Similarly, the p-value for comparing amounts models 6 and 8 lies between 0.000 and 0.048. The evidence here is less compelling, but model 6 is certainly rejected in favor of model 8 at the 5% level.
4.1. Model Checking
 Before attempting to interpret the results of any modeling exercise, it is necessary to carry out thorough checks. For a statistical model, such checks fall broadly into three categories: assessment of predictive ability, checks on probability structure and checks for unexplained systematic structure. The literature on statistical model checking is extensive; relevant overviews are given by McCullagh and Nelder  and Chandler [1998a]. For the GLMs considered here, several simple but informative techniques are available. More details are given by Wheater et al. [2000, chap. 4].
 Throughout this modeling exercise, a variety of simple diagnostics have been used to check models and suggest possible extensions. For example, to check that systematic structure has been captured by a model, we define Pearson residuals for each case in the data set:
where Yi is the observed response for the ith case, and μi and σi are the modeled mean and standard deviation. If the fitted model is correct, all of the Pearson residuals have expectation zero and variance 1. In particular, the mean Pearson residual for any subset of the data should be close to zero, and the root mean squared residual should be close to 1. By appropriate selection of subsets, we can therefore use the residuals to check for unexplained structure. An example is given in Figure 3. The top plots here show the mean and root mean square of Pearson residuals in each year from occurrence model 5 in Table 1, which includes the NAO as a predictor. The dashed lines on the mean plot are approximate 95% confidence bands about zero; if the model is correct, around 95% of mean residuals should lie within these bands. The bands are adjusted for dependence between sites, as described by Wheater et al. [2000, chap. 4]. Their increased width in 1941 and 1997 is due to incomplete records for these years (there are only 22 observations from 1941, and 342 from 1997; recall that on average there are 8.47 observations per day). It is clear from this plot that there is a systematic downward trend in mean residuals between 1940 and 1990. This motivated the addition of a linear trend, and its interactions, to obtain model 6. The annual residual structure for model 6 is shown in the bottom plots of Figure 3. The trend is no longer evident, and by and large the mean residuals lie within the confidence bands. Some lack of fit is evident in the 1950s, which may bear further investigation; apart from this, the only problem is an unusually large mean residual for 1994. No structure is apparent in the root mean square plots.
Figure 3. Annual structure of Pearson residuals from occurrence models 5 (incorporating NAO) and 6 (NAO plus a linear trend). Dotted lines on mean plots show approximate 95% confidence limits under the assumption that the model is correct, adjusted for spatial dependence between sites.
Download figure to PowerPoint
 Pearson residuals are also used to check that seasonal structure is captured by the models (splitting the data set by month) and that regional effects are adequately represented (splitting by site). Seasonality is well represented by all of the models; site-by-site analyses reveal some problems, however. In occurrence model 6, for example, one third of the sites have mean residuals that differ from zero by more than 4 standard errors. However, there does not seem to be any organization in the mean residual pattern; it is therefore likely that the discrepancies here are due to gauge positioning or observer practice, rather than to any deficiency in the model. For example, the mean residuals at sites G3 and G18 are −0.0385 and 0.1559 respectively; the associated standard errors are 0.0082 and 0.0239. Figure 1 shows that the two sites are almost identically located and that their periods of record overlap. A closer examination of the data at these sites reveals that G3 has no trace values, but 17% of wet day values at G18 are traces. It is clear that trace days are being counted as dry at G3 but wet at G18: hence the model, in trying to fit to the average of the two sites, is overpredicting at G3 and underpredicting at G18. Similar explanations can be found for other apparent site-by-site discrepancies.
 As well as checking for systematic residual variation, it is necessary to ensure that the probability structure of the fitted models is correct, since this is used to compute the likelihoods upon which inferences are based. For the amounts model, the simplest check is via quantile-quantile plots of residuals defined in such a way that, if the model is correct, all residuals have the same distribution. The measure used here is the Anscombe residual which, for the gamma distribution, takes the form
If the gamma assumption is correct, all Anscombe residuals have the same distribution which is approximately Gaussian; see, for example, Hougaard . A normal probability plot of Anscombe residuals can therefore be used to test this assumption. For amounts model 8, this plot is shown in Figure 4. The plot shows a good fit except in the lower tail of the distribution, where there are not as many small values as expected under a normal distribution. There are two reasons for this. The first is the presence of trace values, which account for almost all of the points in the lower tail and for which the exact rainfall amounts have been estimated as described in section 3.5 above. The second is that for highly skewed gamma distributions, the Gaussian approximation breaks down in the lower tail since the normal distribution can yield negative values whereas the gamma cannot. To investigate the adequacy of the Gaussian approximation, the dashed line in Figure 4 shows the expected behavior if the gamma assumption is correct. This shows that a substantial part of the discrepancy can be attributed to a breakdown in the approximation. It also shows that the approximation is excellent elsewhere, and reveals some lack of fit in the upper tail of the distribution. However, this discrepancy is slight and there are few data points involved (around 0.6% of the sample), so that for the purposes of our analysis it is not a problem.
Figure 4. Normal probability plot of Anscombe residuals from amounts model 8. The dotted line shows the expected relationship if the gamma assumption is correct.
Download figure to PowerPoint
 For the occurrence model, we cannot use a probability plot to check the forecast probabilities. However, checks can be based on the idea that, if we collect together all of the days when the forecast probability of rain is close to some preassigned value p*, then the overall proportion of these days experiencing rain should be close to p*; see Dawid . For practical implementation, we collect together groups of days for which forecast probabilities are in the intervals (0.0, 0.1),(0.1, 0.2),…,(0.9, 1.0) and compute observed and expected proportions of rainy days within each of these groups (the expected proportion for a subset of M cases with probabilities p1,…,pM is M−1∑i=1Mpi). Unless there is agreement within each forecast decile, there is something wrong with the probability structure of the model. The results, for occurrence model 6, are given in Table 2. This shows good agreement between observed and expected rain day proportions, throughout the range of the forecasts.
Table 2. Observed Versus Expected Proportions of Days With Rain, for Data Grouped According to Forecast Probability of Rainfall Occurrence (Occurrence Model 6)
| ||Forecast Decile|
4.2. Model Interpretation
 Table 1 indicates that the best fitting models are occurrence model 6 and amounts model 8. According to the checks above these both provide a good representation of the structure in the data, and their distributional assumptions are satisfied. The basic predictors in the two models are similar, and are summarized in Table 3. As well as describing the predictors in the model, this shows the maximum likelihood estimates of the cycle lengths and phases for amounts model 8. The nominal standard errors for each of these parameters are small (the highest is 1.60, for the time at which the second cycle reaches its lowest point). The true standard errors will be larger however, as a result of spatial dependence which has not been accounted for here.
Table 3. Summary of Predictors in Best Fitting Occurrence and Amounts Models
|Occurrence Model 6||Amounts Model 8|
|Site effects||site altitude, plus nonparametric Fourier representation using 1 Fourier frequency in each direction (E–W and N–S)||nonparametric polynomial representation, using 3 Legendre polynomials in each direction (E–W and N–S)|
|Interannual variability||NAO, plus linear trend||NAO, plus 2 cycles (lengths 21.8 years and 40.1 years, with minima in 1971 and 1963 respectively)|
|Seasonality||seasonal cosine wave, plus smooth adjustment for December||seasonal cosine wave, plus smooth adjustment for November|
|Autocorrelation||indicators for rain on each of previous 5 days, plus persistence indicators for rain on both previous 2 days and on all previous 7 days||Ln(1 + value x days previously), for x = 1, 2, 3, 4; also trace indicators for each of previous 4 days, and persistence indicators for preceding 3 and 5 days|
|Two-way interactions||autocorrelation with altitude; autocorrelation with interannual variability; autocorrelation with seasonality; seasonality with interannual variability||autocorrelation with interannual variability; autocorrelation with seasonality; seasonality with interannual variability|
|Three-way interactions||NAO with seasonality and autocorrelation||autocorrelation with seasonality and interannual variability|
 Table 3 shows that both models contain a large number of terms representing “autocorrelation” structure, particularly compared to other daily rainfall models in the literature (for example, Stern and Coe  used just 1 previous day's rainfall when modeling rainfall occurrence in West Africa); hence it may appear that our models are unnecessarily complex. However, the primary reason for including these terms is to ensure that within-sequence correlations do not affect inference regarding the effect of other variables upon rainfall. For this purpose it is better to include too many autocorrelation terms than too few. In any case, their inclusion is strongly supported by our analyses. For example, amounts model 8 contains a “persistence indicator” taking the value 1 at any site that has experienced rain on each of the previous 5 days, and zero otherwise. The effect of this indicator varies with the NAO and with the seasonal cycle so that, together with its interactions, it contributes 4 terms to the model. If these terms are dropped, the nominal log likelihood in Table 1 drops by 57.758. The corresponding p-value lies between 0.000 (under independence) and 0.009 (under complete dependence) so that such a reduction is unlikely to arise by chance.
 In each model, seasonal structure is represented by a sine wave, with adjustments for individual months where necessary (i.e., for months with large mean Pearson residuals under a sine-wave-only model). The simplest adjustment is an indicator variable taking the value 1 during the appropriate month, and zero elsewhere. However, a referee has pointed out that this leads to an unnatural model since the resulting seasonal cycle contains discontinuities. We therefore use smooth adjustments based on scaled and shifted bisquare functions:
where d is the day of the month and ℓ is the number of days in the month. These functions decay smoothly to zero at the ends of the month, with a maximum in the middle. The occurrence and amounts models contain adjustments for December and November respectively.
 To visualize the structure of the modeled site effects, Figure 5 maps the surfaces defined by the Fourier and Legendre bases for each of the models. The effect of site altitude in the occurrence model is not included, so that in this case the map shows the regional structure after accounting for altitude. Bearing in mind that the fitted surfaces will be most reliable near gauges, both maps show physically meaningful structures. For the occurrence model the main features are a gentle west-east gradient, and an area of increased rainfall occurrence centered upon the end of Galway Bay. For the amounts model, the pattern is approximately constant except at the western margin, where there are enhanced intensities close to the sea from whence most weather systems arrive. The difference between the two patterns suggests that the primary mechanisms controlling rainfall occurrence and amounts are different.
Figure 5. Spatial variation of rainfall as represented by (left) occurrence model 6 and (right) amounts model 8. For the occurrence model, contours represent contributions to the log odds at equation (2). For the amounts model, contours are multiplicative adjustments to a “baseline” level. Squares are locations of rain gauges (see Figure 1).
Download figure to PowerPoint
 It is of particular interest to try and interpret the interactions in Table 3. Some are easily interpreted: for example, the interactions between seasonality and autocorrelation reflect the fact that temporal dependence in rainfall sequences is stronger in winter than in summer. This in turn has a physical interpretation in terms of the relative frequencies of convective and frontal weather systems: homogeneous frontal systems account for a greater proportion of rainfall in winter than summer.
 The interactions of most interest, however, are those involving the trend functions and the NAO, since these give detailed information about precisely how the rainfall patterns respond to interannual changes. For illustrative purposes, we consider the interaction between the NAO and seasonality in amounts model 8. For this model the contribution to the linear predictor (equation (3)), from terms involving just seasonal effects and the NAO, is
where day is the “day” of the year (running from 1 to 365), fNOV is an adjustment of the form (9) for November, and NAO is the current value of the monthly NAO index.
 If we put NAO = 0 in (10), we obtain an “average” seasonal cycle; by putting NAO = 1 we obtain the corresponding cycle for a year in which NAO takes the value 1 in every month, i.e., in which there is a reasonably strong, and persistent, positive anomaly. (10) represents the contribution to the log mean rainfall: this corresponds to a multiplicative adjustment to the mean rainfall, which is plotted in Figure 6. According to Figure 6, rainfall amounts on wet days are highest, on average, in the autumn. The average effect of an enhanced NAO is to increase rainfall amounts substantially throughout the autumn and winter periods, with little effect in the summer. This agrees with our understanding of the NAO as a phenomenon whose effects are mainly confined to the Northern Hemisphere winter [Hurrell, 1995].
Figure 6. Effect of the NAO upon seasonal rainfall structure, according to amounts model 8. Each line represents a multiplicative adjustment to an overall mean level.
Download figure to PowerPoint
 Other interactions in the models can be studied in a similar way. Broadly speaking, we find that the effects of the deterministic trends in each model are to induce wetter winters and drier summers. Moreover, the 3-way interactions involving the NAO suggest that, as well as increasing autumn and winter rainfall amounts, a positive anomaly is associated with decreased autocorrelation in winter rainfall sequences. A physical interpretation is that positive NAO anomalies are associated with weakened organisation in weather systems. The dynamics of this are unclear, but it may be linked to enhanced convective activity.
 Combining all of these results, we find that the extended period of unusually high NAO values in the 1990s is undoubtedly responsible, to some extent, for the high winter rainfalls in our study area. The NAO does not explain all of the trends in rainfall patterns, however: there are other changes, which we have approximated by linear and cyclical trend functions, that have also tended to increase winter rainfalls.