rm(list=ls())
## load dataset
data(breslow.dat, package = "robust")
names(breslow.dat)
[1] "ID" "Y1" "Y2" "Y3" "Y4" "Base" "Age" "Trt"
[9] "Ysum" "sumY" "Age10" "Base4"
summary(breslow.dat[c(6,7,8,10)])
Base Age Trt sumY
Min. : 6.00 Min. :18.00 placebo :28 Min. : 0.00
1st Qu.: 12.00 1st Qu.:23.00 progabide:31 1st Qu.: 11.50
Median : 22.00 Median :28.00 Median : 16.00
Mean : 31.22 Mean :28.34 Mean : 33.05
3rd Qu.: 41.00 3rd Qu.:32.00 3rd Qu.: 36.00
Max. :151.00 Max. :42.00 Max. :302.00
## look at the response variable in more detail
opar <- par(no.readonly = TRUE)
par(mfrow = c(1,2))
attach(breslow.dat)
The following objects are masked from breslow.dat (pos = 3):
Age, Age10, Base, Base4, ID, sumY, Trt, Y1, Y2, Y3, Y4,
Ysum
The following objects are masked from breslow.dat (pos = 4):
Age, Age10, Base, Base4, ID, sumY, Trt, Y1, Y2, Y3, Y4,
Ysum
hist(sumY, breaks = 20, xlab = "Seizure Count", main = "Distribution of Seizures")
boxplot(sumY ~ Trt, xlab = "Treatment", main = "Group Comparisons")
par(opar)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAnUAAAGECAMAAABnDyZHAAAC+lBMVEUAAAABAQECAgIDAwMEBAQFBQUGBgYHBwcICAgJCQkKCgoLCwsMDAwNDQ0ODg4PDw8QEBARERESEhITExMUFBQVFRUWFhYXFxcYGBgZGRkaGhobGxscHBwdHR0eHh4fHx8gICAhISEiIiIjIyMkJCQlJSUmJiYnJycoKCgpKSkqKiorKyssLCwtLS0uLi4vLy8wMDAxMTEyMjIzMzM0NDQ1NTU2NjY3Nzc4ODg5OTk6Ojo7Ozs8PDw9PT0+Pj4/Pz9AQEBBQUFCQkJDQ0NERERFRUVGRkZHR0dISEhJSUlKSkpLS0tMTExNTU1OTk5PT09QUFBRUVFSUlJTU1NUVFRVVVVWVlZXV1dYWFhZWVlaWlpbW1tcXFxdXV1eXl5fX19gYGBhYWFiYmJjY2NkZGRlZWVmZmZnZ2doaGhpaWlqampra2tsbGxtbW1ubm5vb29wcHBxcXFycnJzc3N0dHR1dXV2dnZ3d3d4eHh5eXl6enp7e3t8fHx9fX1+fn5/f3+AgICBgYGCgoKDg4OEhISFhYWGhoaHh4eIiIiJiYmKioqLi4uMjIyNjY2Ojo6Pj4+QkJCRkZGSkpKTk5OUlJSVlZWWlpaXl5eYmJiZmZmampqbm5ucnJydnZ2enp6fn5+goKChoaGioqKjo6OkpKSlpaWmpqanp6eoqKipqamqqqqrq6usrKytra2urq6vr6+wsLCxsbGysrKzs7O0tLS1tbW2tra3t7e4uLi5ubm6urq7u7u8vLy9vb2+vr6/v7/AwMDBwcHCwsLDw8PExMTFxcXGxsbHx8fIyMjJycnKysrLy8vMzMzNzc3Ozs7Pz8/Q0NDR0dHS0tLT09PU1NTW1tbX19fZ2dna2trb29vc3Nzd3d3e3t7f39/g4ODh4eHi4uLj4+Pk5OTl5eXm5ubn5+fo6Ojp6enq6urr6+vs7Ozt7e3u7u7v7+/w8PDx8fHy8vLz8/P09PT19fX29vb39/f4+Pj5+fn6+vr7+/v8/Pz9/f3+/v7///+FjRfQAAAACXBIWXMAAA7DAAAOwwHHb6hkAAAXlElEQVR4nO3deXgUdbbG8ZMAARKSZlcSNEAA2VHZVGQJ0cs64AjoVcZhiXCHxQURmKsXURgFEpEgjiggDCAECKuExYDDoAIuIDAKAoICCiPbsCaQpH/Pc2vprq6kuypV3V0nOLyfP5Ki63RVa75PL9VJNQkAblTaNwBuQagO+KE64IfqgB+qA36oDvihOuCH6oAfqgN+qA74oTrgh+qAX5iqe5lkNR4ceVT+V1+ifMNRZWUSNTXZmun1/S2pWybes3hp8v01Y5oN+GdoGwSHhbU6iWu38P8hfz1x4mHtH6bVqZP2IjlTnqiaungqQb0VEe8WHUF1N5fwVff7ia8MqkXU+LoQY1q2LPJDnk20SfuHstKoOnWy+PXNfUY06IS62IeozetTHomgqINFRuxtEJwWvuoWS99+bU60zH9tkeoU5tXZs4VolrpUEE3Nb0jf3ybKsLsVYBTe6sQqov5CdKXy0vLmlNtjmv35rBAp8oNeO9GKEq4MjD2hrJSq2/9w5dZTC6W5BGolfV1PNNc7qV4/d+L9cXd2/1jerHTV8wPvrP77Y9oOfSuV66gPoCele1z5+7m+fd+Xvn3/WD1Xx7+6hXqDtnmeA7QsskfvrdIN+244OCTM1V0vQ3d7qlus/pAbXxJPVSe6va/88+1P5K2uejV57QB3kQY8k8rI2cbqBl4SchpVW8nLDa979qdb+ZT0sF4tSakuvyyVefk77236JEYZedRdUnXyrdIN+244OCTM1Yl4inWr1SRS5YXZTxO96X3cbEVlqWaLU57qqOLE9NuJcoo2oE4qI8OJkjNnVKaIvfJV6baJb9QkWu/Zn36l7xFWjJBzqdsv7YAUT34TunP97j8SrVY3eH7Lli05zYiGFq9OvlUndMO6Gw7OCHd10o/1ivJDviI9VP5buIcNfN9XHU0o9CSVpDz/20fUw6i6q5F0V76SVKpy1X8KsYhourq7Iit11eX+b7R6R9XtlPhY2UV+AnX0PuQL8SbR3deKVyffKt2w/oaDM8J/Xxen3tcV1iCq0PPdH+ULtfs6+QHSU11MgbTcmmobVbef6DXpYvdt9KB81RrS8pdEk9XdFVmpq06Iaxte6lpeyq63/JKidlJSUgWqqVX3SSRVlQ8oFruvuy70w/obDs4Ic3U3ytA9nh/ytvrKvU7KcV91yrFcT3VJ8vKj8suAwNVlK/8W4l45TOnJl7T4tVZdkZVFqpNd/EDq7tqL3kOIdNlT3U/VKWKzPFC0OvlW6Yd1NxycEebqVhM97r1rKdz+XKL00+viq05Ox1udS3652EG+D0uQQxUrDe7rbqf7/KsrstJX3d9btvyrstBXen2Qpjwqq5R95koPpq8r/9TvUd20flh3w8EZYT5e14JohfpD3jVy5LfCvSOBYpSWNhavjj4V4pco6iBEA6qaJ8RIb3Ubted1jaSnbjlEg/2rK7LSV90R6ZXnZel7bmOq7F5DNE9a3v/ZF+oG3YOI+hQqc/o9qpvWDetvODgjrO9NDI4nanpD/SHvIWqz5B9zq8hHKuZKdzL5xatLXLe1NdFSIXoQdZ/5FCkNqJPKyJ+ke5tl0svUyD3+1RVZ6avO3Y7ojvGzXm1KNFBcTaD41V/MiJLDlDf4AVHMSul17Naie1Q3rRvW33BwRrjfh62yT3juWp7wXPKhEJ96jxILoVUXpazsKt35bFSWWikNfOo7Sny2kXr9V0SA6vQrdc/rfqrt2WmLS0KsKqcsNvpZ3ef/eFaVKbpHz63yDetvODgjrNVV7/DMT/K/lGquz2tfq1ztbvIbXO4/V/errun6+2ObTVQO7ma2iGk14yOlAXXS897EhHaxd/TYKk/4VadfqX81cW1Gh9pRdbsskN8XE9/87o7oJq+eF37V6ffouVW6Yd0NB2fg9+uAH6oDfqgO+KE64IfqgB+qA36oDvihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+KE64IfqgB+qA36oDvihOuCH6oCfM9XNqOLzuCN7gN8yZ6ob9+p5r80POLIH+C1zqLpp2uIOVAfFoTrgh+qAH6oDfqgO+KE64IfqgB+qA36oDvihOuCH6oCf7ercF8+5SxxCdWDGXnWHx9cvTxRVb9wR8zlUB2ZsVfdVXPywjEULZ45IdO0xHUR1YMZWdZ07XVYX8vqYf/AMqgMztqqLm+ddWu8yHUR1YMZWdW0GeF9IjG5rOojqwIyt6lZEpMzdceDgzvm9IrNMB1EdmLH3GjY7Wf18muRs8zlUB2bsHq87uz8nZ9/ZYhf+o6r3T3MqblcuQHVgJpj3Jm5cKnaBW/vbnNa7lAtQHZixV92GJ/tmusdFUZ0FBu9P3IfqoGS2qltKDbpUGFHhz3OeJINXE6gOLLBVXct+BWIJTZGWBhscOUF1YIGt6iouE+Jf8keni6WVAk+gOrDAVnV3pguRP+UXaWl67cATqA4ssFXdM5VnH5S/5//9tkGBJ1AdWGCruksDqJP8vSk9cDrwBKoDC2wer7vwvfw1c0ehwXpUBxaE+TfYUR1YgOqAH6oDfqgO+KE64IfqgB+qA36oDvihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+KE64BdEdfmHThueIxbVgQV2qjuT+o0QBZOiieqsNBhBdWCBneqO0Roh0mjI6rVDaHPgEVQHFtiuLnGkvDiwfeARVAcW2K0un5RT1y3B3/5D8Gzf1zV4R15MSwg8gurAAnvVxacM7VTtiCj8qMqIwCOoDiywU13uqvThXRuUyxSbqPuFwCOoDiywf7yuIE8c31f0gN32el4V1ioXoDowY7O6/KNXle9XTukvLfjB6x7c10HJbFVX8FpFinr+hrSUbnA9PMKCBbaqmxnxbNbzZQYIVAchsVVdE/kI8QJajeogJPbOEJspf+1bLxfVQShsVXfX8/LXE5WHuFEdhMBWdW9GDFt7TYjllDoc1UHwbFWXP6UyyScmXlGTUB0Ez+bxOvfJXPnb9a1zAq9HdWABfoMd+KE64IfqgB+qA36oDvihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+GnVvXE4HJtDdWCBVp2L7p16JOTNoTqwQKvu+qana1CraT+EtjlUBxbon9cVbHumNrVOOxrC5lAdWFDk1cQv83pHRLjo0RMmV3BfPGd4flhUB5Zo1bn3vNaaKvxu3un8DXelGE0fHl+/PFFUvXFGzwBRHVigVVeLagxeq57EJKOawfBXcfHDMhYtnDki0bUn8ASqAwu06sZ9XuBdPHfcYLhzp8vqQl6fLoEnUB1Y4Hted+qD8+LkNKPgFHHzvEvrXYEnUB1YoFX3RZUyJ8XRGq6vTIbbDPC+kBjdNvAEqgMLtOo6PPij9DW3e2eT4RURKXN3HDi4c36vyKzAE6gOLNCqi1mmfFsRazadnUyK5GyDAVQHFmjV1U9Xvk1LMp8/uz8nZ9/ZYhd+1sorZp1yAaoDM1p1/1dp6XWRnxU71nQ88Blib+zZ7dFip3IBqgMzWnX5qRER1cvQY3kmwzhDLISFrp5DH05Z8K3pMM4QC2GBM8QCP62ecyPaqi8ITIZxhlgIC62evuUHjVeYDOMMsRAWWj2x75Q8jDPEQlho9TTaWPIwzhALYaHVk/ZIgdmcB84QC2GgVbf4gcZj09IlIW0O1YEFWnWJXiFtDtWBBfgrbOCnr+7Gj1dN/hDHElQHFviqW1w/kvYOnhxad6gOLNCqW0L/vY72vheZEdLmUB1YoFXX9I/SP/aKsY1C2hyqAwu06iouU6pbXjGkzaE6sECr7u5nleomtAhpc6gOLNCqmxsx/jv6+IPomSFtDtWBBb4zTrzpIqLoV/AaFhynO1536YusHRdC3ByqAwvw3gTw06rr6hXS5lAdWKBVlyoZ0iOu+eSQNofqwIJij7Dn230Q0uZQHVhQ/Hnd6pYhbQ7VgQXFq1scU+JV8g+dNjy8gurAAq26BYppNQxOhyg7k/qNEAWToonqrDQYQXVgge+cToq4ziYfdnKM1giRRkNWrx1CmwOPoDqwwM7xOqW6RPnv/8XA9oFHUB1YYLe6fFJOXbekUuARVAcWaNVV0ws8q9zXNVD+WjstIfAIqgMLtOrW1bxjzFtjE6u+PVsSePYYxacM7VTtiCj8qMqIwCOoDizQqhvWXv776rz2o4xnc1elD+/aoFym2ETdDX5NANWBBb5POfmb8m1B7ZKuUZAnju8resBup3aG2NiPlAtQHZjRqrtDfQN2Uq0gNpLrPUHs7uY4QyyUTKtuVHRWoSjMqjgopM3hERYs0Kq71oti74ql+0L7vU5UBxbojtd9mTFmWo7pL7CP0Qk8gerAAltnnHixClVN8gg8gerAAntnnDhUpoQTjaE6sMDmGSceQHUQOptnnNhu/oEUqA6swBkngB/OOAH8cMYJ4IczTgA/rbrD53DGCWCiVVfl+XBsDtWBBb5PJa5/NQybQ3VggVbd0uQ6L0zFp5wAB3zKCfBTq9vyc5g2h+rAArU6WiDE1Yk/hL45VAcW+Ko7Q1tC3xyqAwscr25V5Yc0PXJD3wP8B3C8utnRWzSuX0PfA/wHcL66WN/aGqgOZKgO+Hmqi6lWrQrFmpzjxCJUBxao1T2nE9LmUB1YYPvzJtwXz5n8MhSqAwvsVXd4fP3yRFH1xh0xGEB1YIGt6r6Kix+WsWjhzBGJrj2BJ1AdWGCrus6dLqsLeX0MTpqN6sACW9XFzfMurXcFnkB1YIGt6toM8L6QGN028ASqAwtsVbciImXujgMHd87vFZkVeALVgQX2XsNmJ5MiOdtgANWBBXaP153dn5Oz72yxC7/UfqvEtV65ANWBmbB8KvFV7bdKmu5QLkB1YMbxz8JGdeAH1QE/W9UFc4ZYVAd+bFUXzBliUR34sfcIG8QZYlEd+LH5vM7+GWJRHfixWZ39M8SiOvCD17DAD9UBP1QH/FAd8EN1wA/VAT9UB/xQHfBDdcAP1QE/VAf8UB3wQ3XAD9UBP1QH/FAd8EN1wA/VAT9UB/xYq6vQUTshSrdd4d0x/JawVhc5XzshSo854d0x/JbwVndQWxyK6m5hQVSXf+i04bn/UR1YYKe6M6nfCFEwKZqozkqDEVQHFtip7hitESKNhqxeO4Q2Bx5BdWCB7eoSR8qLA9sHHkF1YIHd6vJJOSXxkkr6Fbsf9h4QqayesBjVgRnb93UN3pEX0xL0Ky5qB0Qaf65cgOrAjL3q4lOGdqp2RBR+VGVE4BE8woIFdqrLXZU+vGuDcpliE3W/EHgE1YEF9o/XFeSJ4/uMDtihOrAA700AP1QH/Eqrus6NtV8/eSgjvLcBbnqlVV2jh7SjLX/pGd7bADe9UqvuKW0xG9XdalAd8EN1wA/VAT9UB/xQHfBDdcAP1QE/VAf8UB3wQ3XAD9UBP1QH/FAd8EN1wA/VAT9UB/xQHfBDdcDvJqhubq3+miG+P7Sd67v04ba+5VHhvcEv+bbc/7PwbhoM3QTVveJaoSmTr13c7wXt0odraosLdZsLh7rvaJvu+VZ4Nw2Gbobqavomyuqqy9IW/1RPW7wU7uqOaoujUR0X29W5L54zPD8sqgNL7FV3eHz98kRR9cYdMRhAdWCBreq+iosflrFo4cwRia49gSdQHVhgq7rOnS6rC3l9ugSeQHVgga3q4uZ5l9a79Jfv1Q4+VNuoXDC+hXZJq3K+QxMR3bTFuERtsUl53UQ/bbH23dpiUoy2+Ihuc/f01Ba7PKAtPtrUN9Gyt7bYqaO22Md36/pX6qotNsT5VrjYqq7NAO8LidFt9Zef144+pKt3hj/6joUsSfMtv+FbfHu+trh4esCJtxZqiwsyAk6kL9EW583SFpdP8U1My9QW35utLWZO9U1MWe5b/ld4/pdCiWxVtyIiZe6OAwd3zu8VmVXyNIABe69hs5NJkZzt0M2BW4Ld43Vn9+fk7DvryE2BW0aY35sAsADVAT9UB/xQHfBzprrs1983NPNF43XvjzJZN26G8boprxive/c5k42OznXkfwCYcqa6J9oNM9QrwXjdsMinjdfd2c14Xftmxused5ns0HXAkf8BYMqZ6nTvw/rZmmJyRd37sH56mhwkfPsZ43XfNzTZoe59WGCD6kxWgkNQnclKcAiqM1kJDkF1JivBIajOZCU4BNWZrASHoDqTleAQZ6p7yeRPELZ3NbliTIHxuj4fG6+bPdp43Q9NTHbY4LjJSnCIM9VdMXmfyX3O5Ipmv7l3vtB4Xd7lIDeKXxUsDXj3H/ihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+KE64IfqgB9jdbnHZGeYrxzSXsEZTlSX1c7V5Wv/i1cpp4NKNR4I7MUxwVy5YHrj6GazCoLeKzjJgeqyI5+c28F12O/y9Pg1km+MBwI6XHVMMFeeEvHs0uERrwW7V3CUA9Uld3GLS7XG+10+PKWEgQC2P1iWxgRxZXeVodLXZyrmB7VXcFj4qztLc6Svw+r7reg6VOSbDgTwbXq6el9n88onaa30NYuOBrVXcFj4q9tPO6SvGeX9PgulQaeGEXXTC4wHAksaE8SVcw9ek74+H3U12L2Ck8JfXQ7Jpw5ZRBeLXV5Qruqs7OE0yXDAgFJdUFdeUOa5oPcKTnKiOvlTJRZS8d8Nz8uUP5BnSHSB0YABpbogrnyyHz15Pei9gpPCX90+2il9nRkV+KFsNR0yH/CjPsLavvKKyvVWh7BXcJITryYWSF9H1it++fFN8l/brKXTRgMGlOpsXzmLhuSGsldwkgNHTjr3kh4R64wtfvE+Wi99TU10Gw0YUKqze+XrNVPdIe0VnOTEUeKIF7Ifcfl9qqK7W9X0pYNppeGAAaU6u1f+hIany64Eu1dwkhPviC1vHZe82//iC8PjY+7faDIQmPq8zuaV56ifxkKngt0rOAm/cwL8UB3wQ3XAD9UBP1QH/FAd8EN1wA/VAT9UB/xQHfBDdcAP1QE/VAf8UB3wQ3XAD9UBP1QH/FAd8EN1wA/VAT9UB/xQHfBDdcAP1QE/VAf8UB3wK+3qdvdOqNhkvP5D2S/TQGtXda/o5HK1XWzyueyyVZlB3zZwSilX90W5Nu9mvly1VZ7vomsp06xddxT9V1pGT3rZfGpAn+BvHTiklKvr01A+y9zX8jmX7NpGU+SThY2h703HUN1NqJSra9xP+fbiZukBc9690c3lh8PEdPGt94xMMfKpDlO7ShcunVD1sDYi69hMeWw9U/d9IQomN67URj7rum9+cXrjuP7nRDtpMxdK478MTJRydX8oP837nG56xKjMP9CHSnVXd+3ataPRXbm6irokPXtOG5Fcj5zq28zIsi8ve4JW66tr33vDrKgR4njvlMMF3P9VUIJSru7XnlSu46SvpYfKyy75A0ger69UJ5tc8Vt9RXWu6kYk39MKbSsnyr4l3Vf2aKmfbyS1NrAtHmFvSqX9Glb8vGjgbdT+F/ElfSf9a13kDU91WyMXCn1FLwjdiJA/1mKdto0N0mOxEAsj8nTz8qlgx7dCdTel0q2uMFd+bnZ9caX+YqXn7Jo/qtWdrPG0PKBUNESuSHo89Y1IrtBb6jYW/E3MjZA3kyOt8M2nCVR30yrd6g5TjvI9NUF8Th/vlV1VqrvRvoX84ThqRT26qg+7vhFZ4/uU813nxYwU2cp93WK6VnQe1d2sSvm+rnJv+V7qxj2dxfmK86Wl9wa7lWDGxB5SBuKmCXE+zlORb0S2nKbLC5OkcE+UmSE/r2tWbB7V3axK+XndEmoxed4b95TdJsSE6IkrxkZOVoLZQGOlV7G7zonONZesua+5tyJtROZ+mh6e+vajyjsZI8pNUF/DFp2Xq0tt+MmNUvwPhEBK+9XEtm4J5ZMe2yMtuTOaVWyU4VaCmag+gVsjDiXHtJg32VuRNqLK7FnL1WaOfFwk/7VGMcrxuqLzcnWfJ1X6dyn9t4GR0q4ObkWoDvihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+KE64IfqgB+qA36oDvihOuCH6oAfqgN+qA74oTrgh+qAH6oDfqgO+KE64IfqgB+qA37/D5Pxf7cN8VtcAAAAAElFTkSuQmCC)
## from above figure, we can see the skewed nature of the dependent variable and the possible presence of outliers.
## fit the poisson regression
fit <- glm(sumY ~ Base + Age + Trt, data = breslow.dat, family = poisson())
summary(fit)
Call:
glm(formula = sumY ~ Base + Age + Trt, family = poisson(), data = breslow.dat)
Deviance Residuals:
Min 1Q Median 3Q Max
-6.0569 -2.0433 -0.9397 0.7929 11.0061
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) 1.9488259 0.1356191 14.370 < 2e-16 ***
Base 0.0226517 0.0005093 44.476 < 2e-16 ***
Age 0.0227401 0.0040240 5.651 1.59e-08 ***
Trtprogabide -0.1527009 0.0478051 -3.194 0.0014 **
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for poisson family taken to be 1)
Null deviance: 2122.73 on 58 degrees of freedom
Residual deviance: 559.44 on 55 degrees of freedom
AIC: 850.71
Number of Fisher Scoring iterations: 5
detach(breslow.dat)
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