1 /* PSPP - a program for statistical analysis.
2 Copyright (C) 2005, 2009, 2010, 2011, 2012 Free Software Foundation, Inc.
4 This program is free software: you can redistribute it and/or modify
5 it under the terms of the GNU General Public License as published by
6 the Free Software Foundation, either version 3 of the License, or
7 (at your option) any later version.
9 This program is distributed in the hope that it will be useful,
10 but WITHOUT ANY WARRANTY; without even the implied warranty of
11 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
12 GNU General Public License for more details.
14 You should have received a copy of the GNU General Public License
15 along with this program. If not, see <http://www.gnu.org/licenses/>. */
21 #include <gsl/gsl_cdf.h>
22 #include <gsl/gsl_matrix.h>
24 #include <data/dataset.h>
26 #include "language/command.h"
27 #include "language/lexer/lexer.h"
28 #include "language/lexer/value-parser.h"
29 #include "language/lexer/variable-parser.h"
32 #include "data/casegrouper.h"
33 #include "data/casereader.h"
34 #include "data/dictionary.h"
36 #include "math/covariance.h"
37 #include "math/linreg.h"
38 #include "math/moments.h"
40 #include "libpspp/message.h"
41 #include "libpspp/taint.h"
43 #include "output/tab.h"
46 #define _(msgid) gettext (msgid)
47 #define N_(msgid) msgid
50 #include <gl/intprops.h>
52 #define REG_LARGE_DATA 1000
58 const struct variable **vars;
61 const struct variable **dep_vars;
77 static void run_regression (const struct regression *cmd, struct casereader *input);
82 Transformations for saving predicted values
87 int n_trns; /* Number of transformations. */
88 int trns_id; /* Which trns is this one? */
89 linreg *c; /* Linear model for this trns. */
93 Gets the predicted values.
96 regression_trns_pred_proc (void *t_, struct ccase **c,
97 casenumber case_idx UNUSED)
101 struct reg_trns *trns = t_;
103 union value *output = NULL;
104 const union value *tmp;
106 const struct variable **vars = NULL;
108 assert (trns != NULL);
110 assert (model != NULL);
111 assert (model->depvar != NULL);
112 assert (model->pred != NULL);
114 vars = linreg_get_vars (model);
115 n_vals = linreg_n_coeffs (model);
116 vals = xnmalloc (n_vals, sizeof (*vals));
117 *c = case_unshare (*c);
119 output = case_data_rw (*c, model->pred);
121 for (i = 0; i < n_vals; i++)
123 tmp = case_data (*c, vars[i]);
126 output->f = linreg_predict (model, vals, n_vals);
128 return TRNS_CONTINUE;
135 regression_trns_resid_proc (void *t_, struct ccase **c,
136 casenumber case_idx UNUSED)
140 struct reg_trns *trns = t_;
142 union value *output = NULL;
143 const union value *tmp;
146 const struct variable **vars = NULL;
148 assert (trns != NULL);
150 assert (model != NULL);
151 assert (model->depvar != NULL);
152 assert (model->resid != NULL);
154 vars = linreg_get_vars (model);
155 n_vals = linreg_n_coeffs (model);
157 vals = xnmalloc (n_vals, sizeof (*vals));
158 *c = case_unshare (*c);
159 output = case_data_rw (*c, model->resid);
160 assert (output != NULL);
162 for (i = 0; i < n_vals; i++)
164 tmp = case_data (*c, vars[i]);
167 tmp = case_data (*c, model->depvar);
169 output->f = linreg_residual (model, obs, vals, n_vals);
172 return TRNS_CONTINUE;
177 reg_get_name (const struct dictionary *dict, const char *prefix)
182 /* XXX handle too-long prefixes */
183 name = xmalloc (strlen (prefix) + INT_BUFSIZE_BOUND (i) + 1);
186 sprintf (name, "%s%d", prefix, i);
187 if (dict_lookup_var (dict, name) == NULL)
193 Free the transformation. Free its linear model if this
194 transformation is the last one.
197 regression_trns_free (void *t_)
200 struct reg_trns *t = t_;
202 if (t->trns_id == t->n_trns)
204 result = linreg_free (t->c);
212 reg_save_var (struct dataset *ds, const char *prefix, trns_proc_func * f,
213 linreg * c, struct variable **v, int n_trns)
215 struct dictionary *dict = dataset_dict (ds);
216 static int trns_index = 1;
218 struct variable *new_var;
219 struct reg_trns *t = NULL;
221 t = xmalloc (sizeof (*t));
222 t->trns_id = trns_index;
226 name = reg_get_name (dict, prefix);
227 new_var = dict_create_var_assert (dict, name, 0);
231 add_transformation (ds, f, regression_trns_free, t);
236 subcommand_save (const struct regression *cmd)
241 if ( cmd->resid ) n_trns++;
242 if ( cmd->pred ) n_trns++;
244 n_trns *= cmd->n_dep_vars;
246 for (lc = cmd->models; lc < cmd->models + cmd->n_dep_vars; lc++)
250 if ((*lc)->depvar != NULL)
254 reg_save_var (cmd->ds, "RES", regression_trns_resid_proc, *lc,
255 &(*lc)->resid, n_trns);
259 reg_save_var (cmd->ds, "PRED", regression_trns_pred_proc, *lc,
260 &(*lc)->pred, n_trns);
268 cmd_regression (struct lexer *lexer, struct dataset *ds)
271 struct regression regression;
272 const struct dictionary *dict = dataset_dict (ds);
274 memset (®ression, 0, sizeof (struct regression));
276 regression.anova = true;
277 regression.coeff = true;
280 regression.pred = false;
281 regression.resid = false;
285 /* Accept an optional, completely pointless "/VARIABLES=" */
286 lex_match (lexer, T_SLASH);
287 if (lex_match_id (lexer, "VARIABLES"))
289 if (! lex_force_match (lexer, T_EQUALS) )
293 if (!parse_variables_const (lexer, dict,
294 ®ression.vars, ®ression.n_vars,
295 PV_NO_DUPLICATE | PV_NUMERIC))
299 while (lex_token (lexer) != T_ENDCMD)
301 lex_match (lexer, T_SLASH);
303 if (lex_match_id (lexer, "DEPENDENT"))
305 if (! lex_force_match (lexer, T_EQUALS) )
308 if (!parse_variables_const (lexer, dict,
309 ®ression.dep_vars, ®ression.n_dep_vars,
310 PV_NO_DUPLICATE | PV_NUMERIC))
313 else if (lex_match_id (lexer, "METHOD"))
315 lex_match (lexer, T_EQUALS);
317 if (!lex_force_match_id (lexer, "ENTER"))
322 else if (lex_match_id (lexer, "STATISTICS"))
324 lex_match (lexer, T_EQUALS);
326 while (lex_token (lexer) != T_ENDCMD
327 && lex_token (lexer) != T_SLASH)
329 if (lex_match (lexer, T_ALL))
332 else if (lex_match_id (lexer, "DEFAULTS"))
335 else if (lex_match_id (lexer, "R"))
338 else if (lex_match_id (lexer, "COEFF"))
341 else if (lex_match_id (lexer, "ANOVA"))
344 else if (lex_match_id (lexer, "BCOV"))
349 lex_error (lexer, NULL);
354 else if (lex_match_id (lexer, "SAVE"))
356 lex_match (lexer, T_EQUALS);
358 while (lex_token (lexer) != T_ENDCMD
359 && lex_token (lexer) != T_SLASH)
361 if (lex_match_id (lexer, "PRED"))
363 regression.pred = true;
365 else if (lex_match_id (lexer, "RESID"))
367 regression.resid = true;
371 lex_error (lexer, NULL);
378 lex_error (lexer, NULL);
383 if (!regression.vars)
385 dict_get_vars (dict, ®ression.vars, ®ression.n_vars, 0);
389 regression.models = xcalloc (regression.n_dep_vars, sizeof *regression.models);
392 struct casegrouper *grouper;
393 struct casereader *group;
396 grouper = casegrouper_create_splits (proc_open (ds), dict);
397 while (casegrouper_get_next_group (grouper, &group))
398 run_regression (®ression, group);
399 ok = casegrouper_destroy (grouper);
400 ok = proc_commit (ds) && ok;
403 if (regression.pred || regression.resid )
404 subcommand_save (®ression);
407 for (k = 0; k < regression.n_dep_vars; k++)
408 linreg_free (regression.models[k]);
409 free (regression.models);
410 free (regression.vars);
411 free (regression.dep_vars);
415 for (k = 0; k < regression.n_dep_vars; k++)
416 linreg_free (regression.models[k]);
417 free (regression.models);
418 free (regression.vars);
419 free (regression.dep_vars);
425 get_n_all_vars (const struct regression *cmd)
427 size_t result = cmd->n_vars;
431 result += cmd->n_dep_vars;
432 for (i = 0; i < cmd->n_dep_vars; i++)
434 for (j = 0; j < cmd->n_vars; j++)
436 if (cmd->vars[j] == cmd->dep_vars[i])
446 fill_all_vars (const struct variable **vars, const struct regression *cmd)
452 for (i = 0; i < cmd->n_vars; i++)
454 vars[i] = cmd->vars[i];
456 for (i = 0; i < cmd->n_dep_vars; i++)
459 for (j = 0; j < cmd->n_vars; j++)
461 if (cmd->dep_vars[i] == cmd->vars[j])
469 vars[i + cmd->n_vars] = cmd->dep_vars[i];
475 Is variable k the dependent variable?
478 is_depvar (const struct regression *cmd, size_t k, const struct variable *v)
480 return v == cmd->vars[k];
484 /* Identify the explanatory variables in v_variables. Returns
485 the number of independent variables. */
487 identify_indep_vars (const struct regression *cmd,
488 const struct variable **indep_vars,
489 const struct variable *depvar)
491 int n_indep_vars = 0;
494 for (i = 0; i < cmd->n_vars; i++)
495 if (!is_depvar (cmd, i, depvar))
496 indep_vars[n_indep_vars++] = cmd->vars[i];
497 if ((n_indep_vars < 1) && is_depvar (cmd, 0, depvar))
500 There is only one independent variable, and it is the same
501 as the dependent variable. Print a warning and continue.
504 gettext ("The dependent variable is equal to the independent variable."
505 "The least squares line is therefore Y=X."
506 "Standard errors and related statistics may be meaningless."));
508 indep_vars[0] = cmd->vars[0];
515 fill_covariance (gsl_matrix *cov, struct covariance *all_cov,
516 const struct variable **vars,
517 size_t n_vars, const struct variable *dep_var,
518 const struct variable **all_vars, size_t n_all_vars,
523 size_t dep_subscript;
525 const gsl_matrix *ssizes;
526 const gsl_matrix *mean_matrix;
527 const gsl_matrix *ssize_matrix;
530 gsl_matrix *cm = covariance_calculate_unnormalized (all_cov);
535 rows = xnmalloc (cov->size1 - 1, sizeof (*rows));
537 for (i = 0; i < n_all_vars; i++)
539 for (j = 0; j < n_vars; j++)
541 if (vars[j] == all_vars[i])
546 if (all_vars[i] == dep_var)
551 mean_matrix = covariance_moments (all_cov, MOMENT_MEAN);
552 ssize_matrix = covariance_moments (all_cov, MOMENT_NONE);
553 for (i = 0; i < cov->size1 - 1; i++)
555 means[i] = gsl_matrix_get (mean_matrix, rows[i], 0)
556 / gsl_matrix_get (ssize_matrix, rows[i], 0);
557 for (j = 0; j < cov->size2 - 1; j++)
559 gsl_matrix_set (cov, i, j, gsl_matrix_get (cm, rows[i], rows[j]));
560 gsl_matrix_set (cov, j, i, gsl_matrix_get (cm, rows[j], rows[i]));
563 means[cov->size1 - 1] = gsl_matrix_get (mean_matrix, dep_subscript, 0)
564 / gsl_matrix_get (ssize_matrix, dep_subscript, 0);
565 ssizes = covariance_moments (all_cov, MOMENT_NONE);
566 result = gsl_matrix_get (ssizes, dep_subscript, rows[0]);
567 for (i = 0; i < cov->size1 - 1; i++)
569 gsl_matrix_set (cov, i, cov->size1 - 1,
570 gsl_matrix_get (cm, rows[i], dep_subscript));
571 gsl_matrix_set (cov, cov->size1 - 1, i,
572 gsl_matrix_get (cm, rows[i], dep_subscript));
573 if (result > gsl_matrix_get (ssizes, rows[i], dep_subscript))
575 result = gsl_matrix_get (ssizes, rows[i], dep_subscript);
578 gsl_matrix_set (cov, cov->size1 - 1, cov->size1 - 1,
579 gsl_matrix_get (cm, dep_subscript, dep_subscript));
581 gsl_matrix_free (cm);
587 STATISTICS subcommand output functions.
589 static void reg_stats_r (linreg *, void *);
590 static void reg_stats_coeff (linreg *, void *);
591 static void reg_stats_anova (linreg *, void *);
592 static void reg_stats_bcov (linreg *, void *);
594 static void statistics_keyword_output (void (*)(linreg *, void *),
595 bool, linreg *, void *);
600 subcommand_statistics (const struct regression *cmd , linreg * c, void *aux)
602 statistics_keyword_output (reg_stats_r, cmd->r, c, aux);
603 statistics_keyword_output (reg_stats_anova, cmd->anova, c, aux);
604 statistics_keyword_output (reg_stats_coeff, cmd->coeff, c, aux);
605 statistics_keyword_output (reg_stats_bcov, cmd->bcov, c, aux);
610 run_regression (const struct regression *cmd, struct casereader *input)
617 struct covariance *cov;
618 const struct variable **vars;
619 const struct variable **all_vars;
620 const struct variable *dep_var;
621 struct casereader *reader;
624 linreg **models = cmd->models;
626 n_all_vars = get_n_all_vars (cmd);
627 all_vars = xnmalloc (n_all_vars, sizeof (*all_vars));
628 fill_all_vars (all_vars, cmd);
629 vars = xnmalloc (cmd->n_vars, sizeof (*vars));
630 means = xnmalloc (n_all_vars, sizeof (*means));
631 cov = covariance_1pass_create (n_all_vars, all_vars,
632 dict_get_weight (dataset_dict (cmd->ds)), MV_ANY);
634 reader = casereader_clone (input);
635 reader = casereader_create_filter_missing (reader, all_vars, n_all_vars,
639 for (; (c = casereader_read (reader)) != NULL; case_unref (c))
641 covariance_accumulate (cov, c);
644 for (k = 0; k < cmd->n_dep_vars; k++)
649 dep_var = cmd->dep_vars[k];
650 n_indep = identify_indep_vars (cmd, vars, dep_var);
652 this_cm = gsl_matrix_alloc (n_indep + 1, n_indep + 1);
653 n_data = fill_covariance (this_cm, cov, vars, n_indep,
654 dep_var, all_vars, n_all_vars, means);
655 models[k] = linreg_alloc (dep_var, (const struct variable **) vars,
657 models[k]->depvar = dep_var;
658 for (i = 0; i < n_indep; i++)
660 linreg_set_indep_variable_mean (models[k], i, means[i]);
662 linreg_set_depvar_mean (models[k], means[i]);
664 For large data sets, use QR decomposition.
666 if (n_data > sqrt (n_indep) && n_data > REG_LARGE_DATA)
668 models[k]->method = LINREG_QR;
674 Find the least-squares estimates and other statistics.
676 linreg_fit (this_cm, models[k]);
678 if (!taint_has_tainted_successor (casereader_get_taint (input)))
680 subcommand_statistics (cmd, models[k], this_cm);
686 _("No valid data found. This command was skipped."));
687 linreg_free (models[k]);
690 gsl_matrix_free (this_cm);
693 casereader_destroy (reader);
697 casereader_destroy (input);
698 covariance_destroy (cov);
706 reg_stats_r (linreg *c, void *aux UNUSED)
716 rsq = linreg_ssreg (c) / linreg_sst (c);
717 adjrsq = 1.0 - (1.0 - rsq) * (linreg_n_obs (c) - 1.0) / (linreg_n_obs (c) - linreg_n_coeffs (c));
718 std_error = sqrt (linreg_mse (c));
719 t = tab_create (n_cols, n_rows);
720 tab_box (t, TAL_2, TAL_2, -1, TAL_1, 0, 0, n_cols - 1, n_rows - 1);
721 tab_hline (t, TAL_2, 0, n_cols - 1, 1);
722 tab_vline (t, TAL_2, 2, 0, n_rows - 1);
723 tab_vline (t, TAL_0, 1, 0, 0);
725 tab_text (t, 1, 0, TAB_CENTER | TAT_TITLE, _("R"));
726 tab_text (t, 2, 0, TAB_CENTER | TAT_TITLE, _("R Square"));
727 tab_text (t, 3, 0, TAB_CENTER | TAT_TITLE, _("Adjusted R Square"));
728 tab_text (t, 4, 0, TAB_CENTER | TAT_TITLE, _("Std. Error of the Estimate"));
729 tab_double (t, 1, 1, TAB_RIGHT, sqrt (rsq), NULL);
730 tab_double (t, 2, 1, TAB_RIGHT, rsq, NULL);
731 tab_double (t, 3, 1, TAB_RIGHT, adjrsq, NULL);
732 tab_double (t, 4, 1, TAB_RIGHT, std_error, NULL);
733 tab_title (t, _("Model Summary"));
738 Table showing estimated regression coefficients.
741 reg_stats_coeff (linreg * c, void *aux_)
753 const struct variable *v;
755 gsl_matrix *cov = aux_;
758 n_rows = linreg_n_coeffs (c) + 3;
760 t = tab_create (n_cols, n_rows);
761 tab_headers (t, 2, 0, 1, 0);
762 tab_box (t, TAL_2, TAL_2, -1, TAL_1, 0, 0, n_cols - 1, n_rows - 1);
763 tab_hline (t, TAL_2, 0, n_cols - 1, 1);
764 tab_vline (t, TAL_2, 2, 0, n_rows - 1);
765 tab_vline (t, TAL_0, 1, 0, 0);
767 tab_text (t, 2, 0, TAB_CENTER | TAT_TITLE, _("B"));
768 tab_text (t, 3, 0, TAB_CENTER | TAT_TITLE, _("Std. Error"));
769 tab_text (t, 4, 0, TAB_CENTER | TAT_TITLE, _("Beta"));
770 tab_text (t, 5, 0, TAB_CENTER | TAT_TITLE, _("t"));
771 tab_text (t, 6, 0, TAB_CENTER | TAT_TITLE, _("Significance"));
772 tab_text (t, 1, 1, TAB_LEFT | TAT_TITLE, _("(Constant)"));
773 tab_double (t, 2, 1, 0, linreg_intercept (c), NULL);
774 std_err = sqrt (gsl_matrix_get (linreg_cov (c), 0, 0));
775 tab_double (t, 3, 1, 0, std_err, NULL);
776 tab_double (t, 4, 1, 0, 0.0, NULL);
777 t_stat = linreg_intercept (c) / std_err;
778 tab_double (t, 5, 1, 0, t_stat, NULL);
779 pval = 2 * gsl_cdf_tdist_Q (fabs (t_stat), (double) (linreg_n_obs (c) - linreg_n_coeffs (c)));
780 tab_double (t, 6, 1, 0, pval, NULL);
781 for (j = 0; j < linreg_n_coeffs (c); j++)
784 ds_init_empty (&tstr);
787 v = linreg_indep_var (c, j);
788 label = var_to_string (v);
789 /* Do not overwrite the variable's name. */
790 ds_put_cstr (&tstr, label);
791 tab_text (t, 1, this_row, TAB_CENTER, ds_cstr (&tstr));
793 Regression coefficients.
795 tab_double (t, 2, this_row, 0, linreg_coeff (c, j), NULL);
797 Standard error of the coefficients.
799 std_err = sqrt (gsl_matrix_get (linreg_cov (c), j + 1, j + 1));
800 tab_double (t, 3, this_row, 0, std_err, NULL);
802 Standardized coefficient, i.e., regression coefficient
803 if all variables had unit variance.
805 beta = sqrt (gsl_matrix_get (cov, j, j));
806 beta *= linreg_coeff (c, j) /
807 sqrt (gsl_matrix_get (cov, cov->size1 - 1, cov->size2 - 1));
808 tab_double (t, 4, this_row, 0, beta, NULL);
811 Test statistic for H0: coefficient is 0.
813 t_stat = linreg_coeff (c, j) / std_err;
814 tab_double (t, 5, this_row, 0, t_stat, NULL);
816 P values for the test statistic above.
819 2 * gsl_cdf_tdist_Q (fabs (t_stat),
820 (double) (linreg_n_obs (c) - linreg_n_coeffs (c)));
821 tab_double (t, 6, this_row, 0, pval, NULL);
824 tab_title (t, _("Coefficients"));
829 Display the ANOVA table.
832 reg_stats_anova (linreg * c, void *aux UNUSED)
836 const double msm = linreg_ssreg (c) / linreg_dfmodel (c);
837 const double mse = linreg_mse (c);
838 const double F = msm / mse;
839 const double pval = gsl_cdf_fdist_Q (F, c->dfm, c->dfe);
844 t = tab_create (n_cols, n_rows);
845 tab_headers (t, 2, 0, 1, 0);
847 tab_box (t, TAL_2, TAL_2, -1, TAL_1, 0, 0, n_cols - 1, n_rows - 1);
849 tab_hline (t, TAL_2, 0, n_cols - 1, 1);
850 tab_vline (t, TAL_2, 2, 0, n_rows - 1);
851 tab_vline (t, TAL_0, 1, 0, 0);
853 tab_text (t, 2, 0, TAB_CENTER | TAT_TITLE, _("Sum of Squares"));
854 tab_text (t, 3, 0, TAB_CENTER | TAT_TITLE, _("df"));
855 tab_text (t, 4, 0, TAB_CENTER | TAT_TITLE, _("Mean Square"));
856 tab_text (t, 5, 0, TAB_CENTER | TAT_TITLE, _("F"));
857 tab_text (t, 6, 0, TAB_CENTER | TAT_TITLE, _("Significance"));
859 tab_text (t, 1, 1, TAB_LEFT | TAT_TITLE, _("Regression"));
860 tab_text (t, 1, 2, TAB_LEFT | TAT_TITLE, _("Residual"));
861 tab_text (t, 1, 3, TAB_LEFT | TAT_TITLE, _("Total"));
863 /* Sums of Squares */
864 tab_double (t, 2, 1, 0, linreg_ssreg (c), NULL);
865 tab_double (t, 2, 3, 0, linreg_sst (c), NULL);
866 tab_double (t, 2, 2, 0, linreg_sse (c), NULL);
869 /* Degrees of freedom */
870 tab_text_format (t, 3, 1, TAB_RIGHT, "%g", c->dfm);
871 tab_text_format (t, 3, 2, TAB_RIGHT, "%g", c->dfe);
872 tab_text_format (t, 3, 3, TAB_RIGHT, "%g", c->dft);
875 tab_double (t, 4, 1, TAB_RIGHT, msm, NULL);
876 tab_double (t, 4, 2, TAB_RIGHT, mse, NULL);
878 tab_double (t, 5, 1, 0, F, NULL);
880 tab_double (t, 6, 1, 0, pval, NULL);
882 tab_title (t, _("ANOVA"));
888 reg_stats_bcov (linreg * c, void *aux UNUSED)
900 n_cols = c->n_indeps + 1 + 2;
901 n_rows = 2 * (c->n_indeps + 1);
902 t = tab_create (n_cols, n_rows);
903 tab_headers (t, 2, 0, 1, 0);
904 tab_box (t, TAL_2, TAL_2, -1, TAL_1, 0, 0, n_cols - 1, n_rows - 1);
905 tab_hline (t, TAL_2, 0, n_cols - 1, 1);
906 tab_vline (t, TAL_2, 2, 0, n_rows - 1);
907 tab_vline (t, TAL_0, 1, 0, 0);
908 tab_text (t, 0, 0, TAB_CENTER | TAT_TITLE, _("Model"));
909 tab_text (t, 1, 1, TAB_CENTER | TAT_TITLE, _("Covariances"));
910 for (i = 0; i < linreg_n_coeffs (c); i++)
912 const struct variable *v = linreg_indep_var (c, i);
913 label = var_to_string (v);
914 tab_text (t, 2, i, TAB_CENTER, label);
915 tab_text (t, i + 2, 0, TAB_CENTER, label);
916 for (k = 1; k < linreg_n_coeffs (c); k++)
918 col = (i <= k) ? k : i;
919 row = (i <= k) ? i : k;
920 tab_double (t, k + 2, i, TAB_CENTER,
921 gsl_matrix_get (c->cov, row, col), NULL);
924 tab_title (t, _("Coefficient Correlations"));
929 statistics_keyword_output (void (*function) (linreg *, void *),
930 bool keyword, linreg * c, void *aux)
934 (*function) (c, aux);