5
$\begingroup$

I have the following data frame

structure(list(Chi = structure(c(1L, 1L, 5L, 5L, 6L, 9L, 9L, 
12L, 13L, 14L, 14L, 16L, 16L, 19L, 19L, 20L, 20L, 23L, 24L, 24L, 
26L, 26L, 31L, 31L, 33L, 33L, 36L, 37L, 37L, 40L, 40L, 43L, 43L, 
44L, 44L, 45L, 46L, 47L, 47L, 48L, 48L, 52L, 52L, 54L, 54L, 55L, 
55L, 56L, 59L, 59L, 61L, 61L, 63L, 63L, 64L, 64L, 65L, 65L, 69L, 
69L, 70L, 70L, 71L, 71L, 72L, 72L, 75L, 75L, 76L, 76L, 77L, 77L, 
79L, 79L, 86L, 86L, 87L, 87L, 88L, 88L, 91L, 91L, 92L, 92L, 93L, 
95L, 96L, 96L, 97L, 97L, 98L, 98L, 99L, 99L, 100L, 100L, 101L, 
101L, 103L, 103L, 104L, 104L, 107L, 108L, 108L, 112L, 112L, 113L, 
116L, 116L, 117L, 120L, 125L, 125L, 127L, 127L, 129L, 129L, 130L, 
131L, 131L, 132L, 132L, 134L, 134L, 135L, 135L, 136L, 136L, 139L, 
141L, 141L, 143L, 144L, 144L, 145L, 145L, 146L, 150L, 150L, 151L, 
151L, 153L, 153L, 155L, 155L, 157L, 162L, 162L, 163L, 163L, 164L, 
164L, 167L, 167L, 168L, 169L, 169L, 171L, 171L, 172L, 172L, 174L, 
174L, 175L, 175L, 177L, 177L, 180L, 180L, 183L, 187L, 27L, 83L, 
83L, 165L, 165L, 85L, 85L, 156L, 156L, 17L, 17L, 123L, 123L, 
124L, 124L, 57L, 57L, 42L, 42L, 159L, 159L, 38L, 38L, 82L, 82L, 
41L, 41L, 142L, 142L), .Label = c("0106610856", "0107470802", 
"0108490513", "0108590534", "0109480651", "0111290260", "0111410339", 
"0201390418", "0207570604", "0208360352", "0212323105", "0212380362", 
"0301310432", "0302705635", "0303450495", "0304260266", "0304440574", 
"0305280546", "0305380338", "0305381393", "0305510576", "0305542214", 
"0308610733", "0309370345", "0309665035", "0310380545", "0403320259", 
"0403360374", "0404360343", "0406270198", "0501451137", "0504460676", 
"0511310366", "0605270511", "0605340560", "0605410461", "0605410585", 
"0606260684", "0606270353", "0609360507", "0702520535", "0702570818", 
"0705430421", "0710380364", "0801330378", "0801430275", "0802320430", 
"0803510802", "0805390383", "0806560533", "0809430460", "0902380354", 
"0904340252", "0904370445", "0906340403", "0907380379", "0909415420", 
"0910300100", "0911430253", "1001270460", "1001360389", "1002455294", 
"1005280487", "1006330445", "1009350447", "1010375156", "1011270447", 
"1012350312", "1012400441", "1102570648", "1105450589", "1106230566", 
"1106330587", "1204530475", "1206350342", "1208330373", "1209280345", 
"1209400502", "1209400561", "1210380536", "1302240455", "1305751256", 
"1306370353", "1307260470", "1310340250", "1312430613", "1312440597", 
"1312690593", "1404430512", "1404530479", "1405330376", "1406310360", 
"1406350419", "1406430439", "1408460602", "1412360366", "1502385236", 
"1503370488", "1503470628", "1503660400", "1506390447", "1508340196", 
"1510340688", "1510440453", "1603310622", "1604440376", "1606370014", 
"1609650549", "1610345304", "1610345304x", "1612300367", "1702330397", 
"1704330181", "1706330316", "1712560522", "1802340270", "1804310336", 
"1808430417", "1810400244", "1902340299", "1902610679", "1905360355", 
"1906320438", "1906390525", "1909310514", "1912460408", "2002440204", 
"2004350288", "2007350203", "2009400364", "2009460669", "2011410428", 
"2011500524", "2103335236", "2109370262", "2112290355", "2201330484", 
"2201600686", "2203290471", "2203406259", "2205430513", "2207340473", 
"2208340396", "2303430410", "2303530717", "2308290390", "2309420506", 
"2310370398", "2310370398.0", "2312280310", "2404436295", "2406640663", 
"2411420404", "2501520858", "2505330239", "2505380376", "2511320428", 
"2511320436", "2511360306", "2601490470", "2601520566", "2608450598", 
"2611400237", "2701470625", "2702230407", "2702340342", "2703470916", 
"2704380538", "2709250586", "2712350545", "2712541146", "2805310438", 
"2805350472", "2807360475", "2807480594", "2809325316", "2809470634", 
"2902400411", "2903350442", "2905330376", "2906450480", "2910240363", 
"3004510529", "3007230195", "3012410333", "3107440299", "3108350420"
), class = "factor"), Sex = structure(c(2L, 2L, 2L, 2L, 1L, 1L, 
1L, 1L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 
2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 
2L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 
1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 1L, 
1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 
2L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 
1L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 1L, 2L, 2L, 2L, 
2L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 
2L, 2L, 2L), .Label = c("F", "M"), class = "factor"), Age = c(50L, 
50L, 63L, 63L, 83L, 55L, 55L, 72L, 81L, 42L, 42L, 86L, 86L, 74L, 
74L, 74L, 74L, 50L, 74L, 74L, 73L, 73L, 67L, 67L, 79L, 79L, 71L, 
70L, 70L, 75L, 75L, 68L, 68L, 73L, 73L, 79L, 69L, 79L, 79L, 61L, 
61L, 74L, 74L, 74L, 74L, 77L, 77L, 73L, 68L, 68L, 76L, 76L, 84L, 
84L, 78L, 78L, 77L, 77L, 71L, 71L, 55L, 55L, 67L, 67L, 88L, 88L, 
77L, 77L, 78L, 78L, 84L, 84L, 71L, 71L, 69L, 69L, 67L, 67L, 41L, 
41L, 78L, 78L, 80L, 80L, 76L, 66L, 76L, 76L, 73L, 73L, 74L, 74L, 
64L, 64L, 46L, 46L, 72L, 72L, 78L, 78L, 67L, 67L, 74L, 47L, 47L, 
79L, 79L, 79L, 78L, 78L, 81L, 77L, 79L, 79L, 67L, 67L, 76L, 76L, 
70L, 64L, 64L, 70L, 70L, 79L, 79L, 74L, 74L, 82L, 82L, 83L, 69L, 
69L, 76L, 69L, 69L, 58L, 58L, 83L, 83L, 83L, 68L, 68L, 69L, 69L, 
79L, 79L, 79L, 66L, 66L, 70L, 70L, 65L, 65L, 65L, 65L, 72L, 87L, 
87L, 57L, 57L, 80L, 80L, 76L, 76L, 63L, 63L, 64L, 64L, 78L, 78L, 
60L, 76L, 80L, 75L, 75L, 90L, 90L, 78L, 78L, 74L, 74L, 69L, 69L, 
80L, 80L, 73L, 73L, 71L, 71L, 56L, 56L, 76L, 76L, 87L, 87L, 38L, 
38L, 61L, 61L, 78L, 78L), SBR = c(12.061, 11.447, 9.403, 9.136, 
9.747, 8.648, 7.934, 7.914, 9.349, 11.224, 10.433, 4.897, 5.823, 
8.683, 8.692, 13.018, 13.386, 7.817, 7.384, 7.518, 11.091, 11.028, 
8.372, 8.497, 10.751, 10.488, 4.347, 2.593, 2.203, 6.461, 7.272, 
4.581, 4.593, 10.31, 9.004, 10.362, 10.307, 9.266, 10.163, 9.24, 
8.732, 8.449, 7.823, 10.427, 10.669, 8.695, 8.729, 8.653, 12.299, 
12.158, 11.748, 11.19, 8.431, 8.717, 8.253, 8.412, 6.911, 6.805, 
9.468, 11.413, 6.603, 7.697, 7.762, 7.097, 10.607, 8.162, 5.419, 
5.575, 7.007, 6.974, 8.708, 8.419, 9.47, 8.42, 8.229, 8.027, 
5.294, 4.628, 11.475, 10.328, 7.905, 8.491, 10.724, 9.02, 9.095, 
5.754, 9.805, 7.332, 6.669, 5.118, 12.443, 11.972, 13.309, 13.906, 
14.963, 15.119, 6.465, 6.38, 6.949, 6.064, 6.541, 6.648, 3.542, 
11.148, 11.918, 9.743, 9.795, 6.103, 6.025, 3.917, 7.304, 7.628, 
8.092, 7.347, 9.051, 8.206, 10.697, 10.286, 4.564, 10.62, 9.84, 
9.105, 7.998, 6.437, 5.707, 6.949, 6.315, 6.165, 6.68, 8.86, 
8.326, 8.6, 7.776, 5.193, 5.456, 11.864, 11.381, 6.385, 10.972, 
9.87, 9.645, 7.738, 10.096, 9.667, 9.687, 8.255, 4.606, 8.738, 
8.519, 7.002, 6.288, 10.425, 10.303, 8.278, 8.342, 6.657, 6.111, 
5.928, 13.06, 12.747, 5.545, 5.845, 9.338, 9.534, 9.635, 8.716, 
7.765, 7.254, 7.517, 7.317, 7.335, 5.628, 4.864, 7.1, 7.02, 6.734, 
5.622, 7.167, 7.391, 6.443, 6.874, 8.373, 7.573, 5.701, 6.355, 
6.884, 6.296, 9.097, 9.645, 7.068, 7.252, 6, 5.794, 8.074, 9.267, 
12.584, 10.723, 9.39, 9.165, 9.635, 8.814), Diagnosis = structure(c(2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 
2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L), .Label = c("A", "N"), class = "factor"), 
    fit = c(10.1654358296645, 10.1654358296645, 9.07109655193284, 
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    4.27163348349591, 4.27163348349591, 4.68868944268447, 4.68868944268447, 
    4.68868944268447, 4.68868944268447, 4.10340735387365, 2.81620086292774, 
    2.81620086292774, 5.34559069482765, 5.34559069482765, 3.42248381273475, 
    3.42248381273475, 3.76454793766266, 3.76454793766266, 4.85411038352648, 
    4.85411038352648, 4.77149984958079, 4.77149984958079, 3.59391587315661, 
    3.59391587315661, 5.10074511800645, 3.76454793766266, 3.42248381273475, 
    3.84956354899072, 3.84956354899072, 2.55340019612735, 2.55340019612735, 
    3.59391587315661, 3.59391587315661, 3.93437871118446, 3.93437871118446, 
    4.35544561188301, 4.35544561188301, 3.42248381273475, 3.42248381273475, 
    4.01899333351728, 4.01899333351728, 4.18762073879345, 4.18762073879345, 
    5.42680991723477, 5.42680991723477, 3.76454793766266, 3.76454793766266, 
    2.81620086292774, 2.81620086292774, 6.85553891121225, 6.85553891121225, 
    5.0187327527967, 5.0187327527967, 3.59391587315661, 3.59391587315661
    ), upr = c(14.4208737134878, 14.4208737134878, 13.2880827203392, 
    13.2880827203392, 11.6111556534885, 13.9812403915557, 13.9812403915557, 
    12.5235467500559, 11.7752467916346, 15.1343548219701, 15.1343548219701, 
    11.3665083107223, 11.3665083107223, 12.3558556149815, 12.3558556149815, 
    12.3558556149815, 12.3558556149815, 14.4208737134878, 12.3558556149815, 
    12.3558556149815, 12.4396008815305, 12.4396008815305, 12.9462852897287, 
    12.9462852897287, 11.9401351379537, 11.9401351379537, 12.6076932540179, 
    12.6920403981972, 12.6920403981972, 12.2723108882934, 12.2723108882934, 
    12.8613364829343, 12.8613364829343, 12.4396008815305, 12.4396008815305, 
    11.9401351379537, 12.7765881586919, 11.9401351379537, 11.9401351379537, 
    13.4601801288326, 13.4601801288326, 12.3558556149815, 12.3558556149815, 
    12.3558556149815, 12.3558556149815, 12.1058226630474, 12.1058226630474, 
    12.4396008815305, 12.8613364829343, 12.8613364829343, 12.1889666107397, 
    12.1889666107397, 11.5294083001765, 11.5294083001765, 12.0228788974822, 
    12.0228788974822, 12.1058226630474, 12.1058226630474, 12.6076932540179, 
    12.6076932540179, 13.9812403915557, 13.9812403915557, 12.9462852897287, 
    12.9462852897287, 11.2043986770552, 11.2043986770552, 12.1058226630474, 
    12.1058226630474, 12.0228788974822, 12.0228788974822, 11.5294083001765, 
    11.5294083001765, 12.6076932540179, 12.6076932540179, 12.7765881586919, 
    12.7765881586919, 12.9462852897287, 12.9462852897287, 15.2243975674713, 
    15.2243975674713, 12.0228788974822, 12.0228788974822, 11.8575911801404, 
    11.8575911801404, 12.1889666107397, 13.0314344693091, 12.1889666107397, 
    12.1889666107397, 12.4396008815305, 12.4396008815305, 12.3558556149815, 
    12.3558556149815, 13.2023333654031, 13.2023333654031, 14.7760802807705, 
    14.7760802807705, 12.5235467500559, 12.5235467500559, 12.0228788974822, 
    12.0228788974822, 12.9462852897287, 12.9462852897287, 12.3558556149815, 
    14.6869893464325, 14.6869893464325, 11.9401351379537, 11.9401351379537, 
    11.9401351379537, 12.0228788974822, 12.0228788974822, 11.7752467916346, 
    12.1058226630474, 11.9401351379537, 11.9401351379537, 12.9462852897287, 
    12.9462852897287, 12.1889666107397, 12.1889666107397, 12.6920403981972, 
    13.2023333654031, 13.2023333654031, 12.6920403981972, 12.6920403981972, 
    11.9401351379537, 11.9401351379537, 12.3558556149815, 12.3558556149815, 
    11.6931017121067, 11.6931017121067, 11.6111556534885, 12.7765881586919, 
    12.7765881586919, 12.1889666107397, 12.7765881586919, 12.7765881586919, 
    13.7198188066704, 13.7198188066704, 11.6111556534885, 11.6111556534885, 
    11.6111556534885, 12.8613364829343, 12.8613364829343, 12.7765881586919, 
    12.7765881586919, 11.9401351379537, 11.9401351379537, 11.9401351379537, 
    13.0314344693091, 13.0314344693091, 12.6920403981972, 12.6920403981972, 
    13.1167838834176, 13.1167838834176, 13.1167838834176, 13.1167838834176, 
    12.5235467500559, 11.2853549077749, 11.2853549077749, 13.8067617423288, 
    13.8067617423288, 11.8575911801404, 11.8575911801404, 12.1889666107397, 
    12.1889666107397, 13.2880827203392, 13.2880827203392, 13.2023333654031, 
    13.2023333654031, 12.0228788974822, 12.0228788974822, 13.5465276525046, 
    12.1889666107397, 11.8575911801404, 12.2723108882934, 12.2723108882934, 
    11.0430759079299, 11.0430759079299, 12.0228788974822, 12.0228788974822, 
    12.3558556149815, 12.3558556149815, 12.7765881586919, 12.7765881586919, 
    11.8575911801404, 11.8575911801404, 12.4396008815305, 12.4396008815305, 
    12.6076932540179, 12.6076932540179, 13.8939024088035, 13.8939024088035, 
    12.1889666107397, 12.1889666107397, 11.2853549077749, 11.2853549077749, 
    15.4956514146983, 15.4956514146983, 13.4601801288326, 13.4601801288326, 
    12.0228788974822, 12.0228788974822)), .Names = c("Chi", "Sex", 
"Age", "SBR", "Diagnosis", "fit", "lwr", "upr"), row.names = c(NA, 
201L), class = "data.frame")

I plot SBR v Age for each Sex using ggplot2

p <- ggplot(sbr_with_pred, aes(x=Age, y=SBR)) + geom_point(aes(col=Sex), 
                                                           shape=19, alpha=0.4) + 
            geom_smooth(aes(col=Sex),method = 'lm', se=FALSE,linetype=2) + 
            geom_ribbon(aes(y = fit, ymin = lwr, ymax = upr, fill = 'prediction'), 
                        linetype =2,alpha = 0.1) + 
            scale_fill_manual('Interval', values = c('blue')) + theme_bw() + 
            theme(legend.position = "right") + 
            scale_y_continuous(limits = c(-3,15.5),breaks = c(0,5,10,15)) + 
            scale_color_manual("Sex", values = c('red','blue'))

which gives the following

enter image description here

I can get the equation of each regression fit easy enough

lm(formula = SBR ~ Age, data = subset(sbr_with_pred, Sex == "F"))
lm(formula = SBR ~ Age, data = subset(sbr_with_pred, Sex == "M"))

However how do I test whether or not they are significantly different (which they are not). I think analysis of covariance is the appropriate test but I do not know how to implement this in R

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2
  • $\begingroup$ "analysis of covariance" is not a test. $\endgroup$
    – PAC
    Aug 1, 2013 at 14:58
  • $\begingroup$ For wannabe econometricians who ended up here, you may find this thread useful! stats.stackexchange.com/questions/33013/… $\endgroup$
    – lomper
    Dec 20, 2021 at 10:57

3 Answers 3

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You have to use a Chow test (wikipedia). It's an application of the Fisher test to test the equality of coefficients among two groups of individuals. You can compute it easily using the sum of squared residuals of each model.

See my gist file to see how I compute the Chow test. In your case, the null hypothesis of equality of coefficients among the two groups cannot be rejected.

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  • 2
    $\begingroup$ The gist file link in this answer is no longer accessible. Is it possible to have the code for the answer here? $\endgroup$
    – lomper
    Dec 20, 2021 at 10:49
  • $\begingroup$ @lomper I saw this related link for Chow test ( link). $\endgroup$
    – Mohamed Rahouma
    Jun 29, 2022 at 17:27
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You would do best to test for a difference in slopes by including sex and a sex:Age interaction in a multiple regression analysis. The t-test of the interaction term will assess whether or not the slopes differ significantly. The R code for your situation would be (I'm guessing): 

lm(formula = SBR ~ Sex + Age + Sex:Age, data = sbr_with_pred)
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  • 2
    $\begingroup$ For clarification, the difference between this solution and mine (see below), is that I test the equality of all coefficients (intercept + slope) whereas @gung test only equality in slope. $\endgroup$
    – PAC
    Aug 1, 2013 at 15:24
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In R you can use anova for an analysis of covariance. I tried quickly with the anova command to run a test with your data but the sample size for the two models are different which gives problems at the moment. Code by PAC also works nicely. Based on gung's answer you can also do an anova test using the following code (also guessing):

library(car)
Anova(lm(SBR~Age*Sex,data=sbr_with_pred))
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